I heard in college that the first popes basically used this to claim power over the christian world:
The Donatio Constantini, which proved to be a falsification in the early Renaissance. It was supposedly a document written by Constantin himself, saying that the patriarch of Rome was head of the church. This is my main problem, after it proved to be false, how could the pope keep his position?
The claim that Rome was the city of Petrus, and thereby the most important patriarchy. Patriarchs became patriarchs because of a certain arrogance to begin with: Apostels built their first churches and they were their bishops: For them a reason to perceive themselves as higher than the ''average mr. bishop''. This is not justified by christian belief, but simply by being their church more special than others. The whole ''patriarch concept'' is not built on religion in that sense, but just people feeling overly proud/arrogant about being bishop of an apostolic church. I mean with this: It does not give a reason to claim power, that's not biblical nor very logical.
The claim that the pope is the representant of Jesus on earth. This is anti the whole ideal of humbleness.
Throughout the Middle Ages popes gained more and more power by not obeying to concilies and democratic decisions by bishops. This reeks of opportunism, instead of a divine right.
I read the bible, and I did not find anything that states there should be a pope. Therefore I strongly doubt the religious part of the claim that he represents Jesus.
I may have made some historical mistakes, as the part about this matter was not very long and I could not find much about this on the internet. My main problem is: What argument did the pope use after it was found that the Donatio Constantini was fake? That was the argument to even be in that position in the first place, so what's the pope's legitimacy after the Donatio Constantini was denounced as untrue? As I've explained I don't see much arguments in favor of the pope in religion or history, the basis of ''the pope concept'' seems to be a combination of opportunism and kings/leaders at the end of the Roman Empire who wanted an emperor, even more, they initially expected that nothing would change in the sense of Europe having an emperor, that was in the end period of the Empire not a thing that even occured to them.