You have pointed to the third argument of the article as legitimately calling into question Apostolic succession in the Catholic Church: to quote from the article the earliest bishops in certain places are simply unknown, and the lists produced in the third and fourth centuries that purported to identify every bishop back to the founding of the church in a particular area were often historically unreliable.
The Apostles and the their successors were busy teaching and writing and baptizing--they were trying to win the world for Christ through the Gospel. So it is no surprise that the early Church didn't have many chronicles written by what we call historians. Lists weren't important--although we do have the complete list when it comes to St. Peter and his successors in Rome.
At any rate, apostolic succession is evident in the very writings of St. Paul to St. Timothy (who received his ministry as Bishop by the "laying on of hands"--St. Paul ordained him 2 Tim 1:6). "Therefore, my child, be strengthened in the grace which is in Christ Jesus; and the things that thou hast heard from me through many witnesses, commend to trustworthy men who shall be competent in turn to teach others." (2 Timothy 2:2)
St. Clement, the third to succeed St. Peter as Bishop of Rome writes:
"Through countryside and city [the apostles] preached, and they appointed their earliest converts, testing them by the Spirit, to be the bishops and deacons of future believers. Nor was this a novelty, for bishops and deacons had been written about a long time earlier. . . . Our apostles knew through our Lord Jesus Christ that there would be strife for the office of bishop. For this reason, therefore, having received perfect foreknowledge, they appointed those who have already been mentioned and afterwards added the further provision that, if they should die, other approved men should succeed to their ministry" (Letter to the Corinthians 42:4–5, 44:1–3 [A.D. 80]).
It is noteworthy that in the early Church, when St. John the Evangelist was still alive, the Church in Corinth (which had its own Bishop) appealed to the Successor of St. Peter—Pope St. Clement—to settle a dispute. Apostolic succession and authority.
"Blessed be the God and Father of our Lord Jesus Christ, who has blessed us with every spiritual blessing on high in Christ." (Eph. 1:3).
Please pray for me, and be assured of mine. God bless...
Php 4:3 - Yes, and I ask you, loyal yokefellow, help these women who have contended at my side in the cause of the gospel, along with Clement and the rest of my fellow workers, whose names are in the book of life.
Clement had been a companion of Paul and the Corinthians were familiar with him through the word of Paul as a man of faith. Clement was a Roman in Philippi, and he had the commendation of Paul, so it is no wonder that the Corinthian church founded by Paul would seek the advice of a man with Pauline approval. Philippi sent help via Epaphroditus while Paul was imprisoned in Rome, so the pattern of appealing to what was Pauline was evident in the churches Paul had founded. So with the Corinthians. It would be natural to inquire of a trusted friend of Paul for advice.