A major problem for Catholicism is that they chose the wrong Apostle to base the doctrine of the Papacy on. Peter, as we have previously established, was not a Pope. He wasn't given authority the other Apostles didn't share, he had a wife, he was the Apostle to the Jews, he was even openly opposed on a matter of faith by Paul.
Compare this with Paul. Paul demonstrably remained celibate. Paul is the Apostle to the gentiles. Paul wrote the most scripture. Paul even wrote the epistle to the Romans, which is not only directed to the Roman Church, but is so packed full of theology, it is often suggested that you could get a near complete view of the Christian faith just by reading it. But what makes Paul so devastating to the Papacy is that he claimed to be behind the most eminent Apostle in nothing.
First Vatican Council (1869-1870) made the following pronouncement about Peter: "At open variance with this clear doctrine of Holy Scripture, as it has ever been understood by the Catholic Church, are the perverse opinions of those who, while they distort the form of government established by Christ the Lord in His Church, deny that Peter, in his single person, preferably to all the other Apostles, whether taken separately or together, was endowed by Christ with a true and proper primacy of jurisdiction; or of those who assert that the same primacy was not bestowed immediately and directly upon Blessed Peter himself, but upon the Church, and through the Church on Peter as her Minister.
If anyone, therefore, shall say that Blessed Peter the Apostle was not appointed the Prince of all the Apostles and the visible Head of the whole Church Militant; or that the same, directly and immediately, received from the same, Our Lord Jesus Christ, a primacy of honor only, and not of true and proper jurisdiction; let him be anathema."
These are some very strong claims. Suffice to say, the Catholic Church considers Peter the most eminent Apostle. Yet, Paul makes two very bold claims: "I consider that I am not at all inferior" and "in nothing was I behind" the most eminent Apostles. In other words, Paul, "at open variance" with the dogmas of the modern Catholic Church, believed he was Peter's equal.
Of course, no Christian worth his salt can believe the claims of First Vatican Council. Even many Catholic scholars left the Church, knowing full well how bonkers these claims were. But let's suppose, for a moment, Peter really is all those things. If that is indeed the case, then Paul, who was equal to Peter, had primacy over the other Apostles, separately or as a whole, as the prince of Apostles, and the visible Head of the whole Church Militant.
Of course, already, this is blasphemy, as Christ is the head of the Church (Ephesians 1:22; 5:23; Colossians 1:18). But ultimately, if Peter had any authority, any honor, Paul likewise had it. If any Apostle was a Pope, Paul was a Pope. Of course, in reality, no Apostle was Pope. The Papacy is a made up doctrine, invented and defended by a false Church. The Pope, far from being the legitimate successor of any Apostle, is a usurper, a blasphemer, a false teacher, and, barring conversion before death, a Hell-bound sinner, who will drag his flock down with him.