There’s no where in the Bible where writers make any claim to Catholicism or even unity under one leader.
Maybe not the Capital 'C' catholicism, but the non-Dispensational believers recognize a universal church made up of the Church militant and triumphant. Not satisfied with that, Rome added the "Church Padecent" because a loved one died, the Vatican wanted money, and buying candles has long been a recognized form of bringing about one's salvation.
Your point is strongest regarding "under one leader". Rome likes to claim that while Jesus Christ was head of the Church, that in absence of His physical presence, that through the powers of tranformative language worked out on Matthew 16:18, this somehow relates to Peter being the first Pope and through a screwy lottery system comes a successor.
I have seen a gallery of arguments against this, mostly challenges to the language, but it is Peter himself, who seems to cancel out the idea that he is "the" (definite article) "rock" when he counts himself as one of many "living stones". (1 Peter 2:5)
Furthermore, we have this passage in Ephesians:
Eph 2:20-22 "...having been built on the foundation of the apostles and prophets, Jesus Christ Himself being the chief cornerstone, in whom the whole building, being fitted together, grows into a holy temple in the Lord, in whom you also are being built together for a dwelling place of God in the Spirit.
Here Paul identified "The" (definite article) "Rock" as Jesus Christ, who surely not physically present on earth when Paul wrote this still retains the identity of not only being the Foundation stone, but also the Corner stone. IOW, it is obvious that Jesus Christ never relinquished His role as head of the Church, never surrendering it to another man or a series of carnal men. Second, Paul doesn't identify Peter as being "the" stone, rather, he counts the apostles and the OT Prophets as the foundation. This theme is picked up again routinely in Revelation (4:4,10; 5:8,14;11:16...) as the foundation is the twenty-four apostles and prophets (or tribes if you wish)
Therefore, Rome is not only at odds with Peter, but with Paul and the LORD who provided John the vision. Those who have been appointed by God were able to demonstrate that authority through miracles. Rome has no such evidence that any of the Popes are legitimate heirs because, oddly enough for a cult that demands "works" - they have no works in what is arguably the most important role to exhibit works. You are correct in your observations that Rome has supplanted Jesus Christ and His testimony in the Scriptures by conjuring up Traditions to replace God's revealed will for what is clearly man's will.
For Rome to say that they alone have authority, their own Sola Potestate, and to derive that from having "Holy Tradition" trump Scriptures, and to declare that Holy Tradition supersedes Scripture is worse than grading your own tests, its writing your own tests and then grading them.
Likewise, Rome is at odds with Peter. He also states that Jesus is the rock on which the church is built.
1 Peter 2
4As you come to him, a living stone rejected by men but in the sight of God chosen and precious, 5 you yourselves like living stones are being built up as a spiritual house, to be a holy priesthood, to offer spiritual sacrifices acceptable to God through Jesus Christ. 6For it stands in Scripture:
“Behold, I am laying in Zion a stone,
a cornerstone chosen and precious,
and whoever believes in him will not be put to shame.”
7So the honor is for you who believe, but for those who do not believe,
“The stone that the builders rejected
has become the cornerstone,”
8and
“A stone of stumbling,
and a rock of offense.”
They stumble because they disobey the word, as they were destined to do.
That matches the Catohlic view of the Church as a collection of hierarchies starting on the family level, where the man of the household is priest, up to the parish level with ministerial preisthood, up to the bishop sovereign in his local Church, and then up to the Pope. On every level the Church Local is a Church Universal scaled down.
It is simply not true that "Rome alone has authority" as you say.