The papacy and its unique authority can be traced back to Jesus.
You may notice some similarity in these verses.
Isaiah 22:22The "key of David" was an oversized key that the vice-regent of the House of David kept in a pouch worn around his neck. The key represented the vice-regent's plenary authority in the king's absence.I will place on his shoulder the key to the house of David; what he opens no one can shut, and what he shuts no one can open.
Rev. 3:7
"These are the words of him who is holy and true, who holds the key of David. What he opens no one can shut, and what he shuts no one can open."
Matthew 16:19
"I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven."
The key and the office that it represents is referenced in Isaiah 22, where we see the key and office being transferred from Shebna to Eliakim.
Isaiah 22In Rev. 3:7, we see that Jesus, He "who is holy and true," is the King of the eternal Davidic kingdom, since he "holds the key of David."“Go, say to this steward,
to Shebna the palace administrator:
What are you doing here and who gave you permission
to cut out a grave for yourself here,
hewing your grave on the height
and chiseling your resting place in the rock?
“Beware, the Lord is about to take firm hold of you
and hurl you away, you mighty man.
He will roll you up tightly like a ball
and throw you into a large country.
There you will die
and there the chariots you were so proud of
will become a disgrace to your master’s house.
I will depose you from your office,
and you will be ousted from your position.
“In that day I will summon my servant, Eliakim son of Hilkiah. I will clothe him with your robe and fasten your sash around him and hand your authority over to him. He will be a father to those who live in Jerusalem and to the people of Judah. I will place on his shoulder the key to the house of David; what he opens no one can shut, and what he shuts no one can open.
In Matthew 16:19, we see Jesus giving the Key(s) of the Kingdom to Peter. Peter becomes the representative or "vicar" of Christ, in the King's earthly absence.
Just as Eliakim became "a father to those who live in Jerusalem and to the people of Judah," Peter became a father (papa/pope) to the people of God.
We know that Peter's unique position as "vicar of Christ" is an "office" because of Isaiah 22, Matthew 16:19, Acts 1:15-26, and the historical record.
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St. Irenaeus"The blessed apostles [Peter and Paul], having founded and built up the church [of Rome] . . . handed over the office of the episcopate to Linus" (Against Heresies 3:3:3 [A.D. 189]).
Tertullian
"[T]his is the way in which the apostolic churches transmit their lists: like the church of the Smyrneans, which records that Polycarp was placed there by John, like the church of the Romans, where Clement was ordained by Peter" (Demurrer Against the Heretics 32:2 [A.D. 200]).
The Little Labyrinth
"Victor . . . was the thirteenth bishop of Rome from Peter" (The Little Labyrinth [A.D. 211], in Eusebius, Church History 5:28:3).
Cyprian of Carthage
"The Lord says to Peter: ‘I say to you,’ he says, ‘that you are Peter, and upon this rock I will build my Church, and the gates of hell will not overcome it. . . . ’ [Matt. 16:18]. On him [Peter] he builds the Church, and to him he gives the command to feed the sheep [John 21:17], and although he assigns a like power to all the apostles, yet he founded a single chair [cathedra], and he established by his own authority a source and an intrinsic reason for that unity. . . . If someone [today] does not hold fast to this unity of Peter, can he imagine that he still holds the faith? If he [should] desert the chair of Peter upon whom the Church was built, can he still be confident that he is in the Church?" (The Unity of the Catholic Church 4; first edition [A.D. 251]).
"Cornelius was made bishop by the decision of God and of his Christ, by the testimony of almost all the clergy, by the applause of the people then present, by the college of venerable priests and good men, at a time when no one had been made [bishop] before him—when the place of [Pope] Fabian, which is the place of Peter, the dignity of the sacerdotal chair, was vacant. Since it has been occupied both at the will of God and with the ratified consent of all of us, whoever now wishes to become bishop must do so outside. For he cannot have ecclesiastical rank who does not hold to the unity of the Church" (Letters 55:[52]):8 [A.D. 253]).
"With a false bishop appointed for themselves by heretics, they dare even to set sail and carry letters from schismatics and blasphemers to the chair of Peter and to the principal church [at Rome], in which sacerdotal unity has its source" (ibid., 59:14).
Eusebius of Caesarea
"Paul testifies that Crescens was sent to Gaul [2 Tim. 4:10], but Linus, whom he mentions in the Second Epistle to Timothy [2 Tim. 4:21] as his companion at Rome, was Peter’s successor in the episcopate of the church there, as has already been shown. Clement also, who was appointed third bishop of the church at Rome, was, as Paul testifies, his co-laborer and fellow-soldier [Phil. 4:3]" (Church History 3:4:9–10 [A.D. 312]).
Pope Julius I
"[The] judgment [against Athanasius] ought to have been made, not as it was, but according to the ecclesiastical canon. . . . Are you ignorant that the custom has been to write first to us and then for a just decision to be passed from this place [Rome]? If, then, any such suspicion rested upon the bishop there [Athanasius of Alexandria], notice of it ought to have been written to the church here. But now, after having done as they pleased, they want to obtain our concurrence, although we never condemned him. Not thus are the constitutions of Paul, not thus the traditions of the Fathers. This is another form of procedure, and a novel practice. . . . What I write about this is for the common good. For what we have heard from the blessed apostle Peter, these things I signify to you" (Letter on Behalf of Athanasius [A.D. 341], contained in Athanasius, Apology Against the Arians 20–35).
Council of Sardica
"[I]f any bishop loses the judgment in some case [decided by his fellow bishops] and still believes that he has not a bad but a good case, in order that the case may be judged anew . . . let us honor the memory of the apostle Peter by having those who have given the judgment write to Julius, bishop of Rome, so that if it seem proper he may himself send arbiters and the judgment may be made again by the bishops of a neighboring province" (Canon 3 [A.D. 342]).
Optatus
"You cannot deny that you are aware that in the city of Rome the episcopal chair was given first to Peter; the chair in which Peter sat, the same who was head—that is why he is also called Cephas [‘Rock’]—of all the apostles; the one chair in which unity is maintained by all" (The Schism of the Donatists 2:2 [A.D. 367]).
Epiphanius of Salamis
"At Rome the first apostles and bishops were Peter and Paul, then Linus, then Cletus, then Clement, the contemporary of Peter and Paul" (Medicine Chest Against All Heresies 27:6 [A.D. 375]).
Pope Damasus I
"Likewise it is decreed: . . . [W]e have considered that it ought to be announced that . . . the holy Roman Church has been placed at the forefront not by the conciliar decisions of other churches, but has received the primacy by the evangelic voice of our Lord and Savior, who says: ‘You are Peter, and upon this rock I will build my Church, and the gates of hell will not prevail against it; and I will give to you the keys of the kingdom of heaven, and whatever you shall have bound on earth will be bound in heaven, and whatever you shall have loosed on earth shall be loosed in heaven’ [Matt. 16:18–19]. The first see [today], therefore, is that of Peter the apostle, that of the Roman Church, which has neither stain nor blemish nor anything like it" (Decree of Damasus 3 [A.D. 382]).
St. Jerome
"[Pope] Stephen . . . was the blessed Peter’s twenty-second successor in the See of Rome" (Against the Luciferians 23 [A.D. 383]).
"Clement, of whom the apostle Paul writing to the Philippians says ‘With Clement and others of my fellow-workers whose names are written in the book of life,’ the fourth bishop of Rome after Peter, if indeed the second was Linus and the third Anacletus, although most of the Latins think that Clement was second after the apostle" (Lives of Illustrious Men 15 [A.D. 396]).
"Since the East, shattered as it is by the long-standing feuds, subsisting between its peoples, is bit by bit tearing into shreds the seamless vest of the Lord . . . I think it my duty to consult the chair of Peter, and to turn to a church [Rome] whose faith has been praised by Paul [Rom. 1:8]. I appeal for spiritual food to the church whence I have received the garb of Christ. . . . Evil children have squandered their patrimony; you alone keep your heritage intact" (Letters 15:1 [A.D. 396]).
... "I follow no leader but Christ and join in communion with none but your blessedness [Pope Damasus I], that is, with the chair of Peter. I know that this is the rock on which the Church has been built. Whoever eats the Lamb outside this house is profane. Anyone who is not in the ark of Noah will perish when the flood prevails" (ibid., 15:2).
"The church here is split into three parts, each eager to seize me for its own. . . . Meanwhile I keep crying, ‘He that is joined to the chair of Peter is accepted by me!’ . . . Therefore, I implore your blessedness [Pope Damasus I] . . . tell me by letter with whom it is that I should communicate in Syria" (ibid., 16:2).
Ambrose of Milan
"[T]hey [the Novatian heretics] have not the succession of Peter, who hold not the chair of Peter, which they rend by wicked schism; and this, too, they do, wickedly denying that sins can be forgiven [by the sacrament of confession] even in the Church, whereas it was said to Peter: ‘I will give unto thee the keys of the kingdom of heaven, and whatsoever thou shalt bind on earth shall be bound also in heaven, and whatsoever thou shall loose on earth shall be loosed also in heaven’[Matt. 16:19]" (Penance 1:7:33 [A.D. 388]).
St. Augustine
"If all men throughout the world were such as you most vainly accuse them of having been, what has the chair of the Roman church done to you, in which Peter sat, and in which Anastasius sits today?" (Against the Letters of Petilani 2:118 [A.D. 402]).
"If the very order of episcopal succession is to be considered, how much more surely, truly, and safely do we number them from Peter himself, to whom, as to one representing the whole Church, the Lord said, ‘Upon this rock I will build my Church’ . . . [Matt. 16:18]. Peter was succeeded by Linus, Linus by Clement, Clement by Anacletus, Anacletus by Evaristus . . . " (Letters 53:1:2 [A.D. 412]).
Council of Ephesus
"Philip the presbyter and legate of the Apostolic See said: ‘There is no doubt, and in fact it has been known in all ages, that the holy and most blessed Peter, prince and head of the apostles, pillar of the faith, and foundation of the Catholic Church, received the keys of the kingdom from our Lord Jesus Christ, the Savior and Redeemer of the human race, and that to him was given the power of loosing and binding sins: who down even to today and forever both lives and judges in his successors. The holy and most blessed pope Celestine, according to due order, is his successor and holds his place, and us he sent to supply his place in this holy synod’" (Acts of the Council, session 3 [A.D. 431]).
Pope Leo I
"As for the resolution of the bishops which is contrary to the Nicene decree, in union with your faithful piety, I declare it to be invalid and annul it by the authority of the holy apostle Peter" (Letters 110 [A.D. 445]).
"Whereupon the blessed Peter, as inspired by God, and about to benefit all nations by his confession, said, ‘You are the Christ, the Son of the living God.’ Not undeservedly, therefore, was he pronounced blessed by the Lord, and derived from the original Rock that solidity which belonged both to his virtue and to his name [Peter]" (The Tome of Leo [A.D. 449]).
Peter Chrysologus
"We exhort you in every respect, honorable brother, to heed obediently what has been written by the most blessed pope of the city of Rome, for blessed Peter, who lives and presides in his own see, provides the truth of faith to those who seek it. For we, by reason of our pursuit of peace and faith, cannot try cases on the faith without the consent of the bishop of Rome" (Letters 25:2 [A.D. 449]).
Council of Chalcedon
"After the reading of the foregoing epistle [The Tome of Leo], the most reverend bishops cried out: ‘This is the faith of the fathers! This is the faith of the apostles! So we all believe! Thus the orthodox believe! Anathema to him who does not thus believe! Peter has spoken thus through Leo! . . . This is the true faith! Those of us who are orthodox thus believe! This is the faith of the Fathers!’" (Acts of the Council, session 2 [A.D. 451]).
The papal infallibility portion was not part of what those of Rome believed, just as Salmon laid it out -- and showed proofs for in his own day, which covered the before and after 1870
There are plenty of OTHER evidences, including partristics which dispute that things are the way you seem to be attempting portray them.
In fact, your whole presentation here is rather intellectually dishonest.
Even in the Church of Rome when the dogmatic proclamation came in 1870 -- men such as Dollinger, one of the most preeminent historians and authorities on early partristics of his time -- headed for the exits.
The cherry-picked stuff which you obviously must have lifted copy/paste from some Romanist apologetics page, doesn't always do as it may appear to do in the eyes of those who already have their minds made up that even singular "papacy" for the church at Rome was existent the beginning (not to mention "infallibility").
According to quite many modern-day, respectable (in scholarly sense) Roman Catholic historians, there was no such thing in the first few centuries as Papacy, as that eventually developed into being.
I could pick apart your presentation, going through it in detail, such as the citing of Matthew 16 ignores Matthew chapter 18 --- which left things to be that in the earliest centuries no one in the early church saw anything like Papal Supremacy singularly for the bishop of Rome, (well, other than a few of those at Rome who made early grabs for it...eventually getting it).
To this day, those of the East are able to make the case that they never accepted the concept --so cannot be faulted for not accepting it now.
Grabbing at some of the rest of it rather at random---
Things like the cite from Peter Chrysologus
neglects to mention that he was under the patriarchate of the bishop of Rome. It stands also to be noted that the notion of patriarchates itself was something of a development.
I notice too that the quotations from Cyprian are shot through with ellipses.
In other instances that man can be plainly read to have been opposed to the "bishop of bishops" idea be attributed to Rome, or anywhere else. And he was not alone in that. Even a Pope (Gregory) was against the very idea in one of his own early letters, but can be mistakenly seen as supporting the bishop of bishops, Universal bishop idea years later, when what he is doing in those letters chiefly, is writing to those of his own patriarchate.
The phrase attributed to Cyprian "where sarcedotal unity has it's source" does not mean that such unity is still reliant singularly upon that source or geographical location, but as Cyprian mentioned when speaking of the boldness of those he perceived as heretics -- they would be so bold as to go even to that large city, instead of hanging around in far-off locations --distant from what was then the old center of Roman Empire, and had a history of being the most noteworthy one of more than a few 'central locations'. That does not mean or guarantee that such concepts as infallibility were then -- or more importantly -- would into perpetuity be attributed to the Church at Rome.
Singular doctrinal perfection and true authority is not a thing inheritable simply for the sake of being in office, particularly when claim to that authority is portrayed as unquestionable and unassailable -- or in other words -- infallible.