That Peter was a leader among the apostles is undeniable. What is clear though is that he was never an "Overseer" over the other apostles and Jesus made this quite clear right after he charged Peter to "Feed his sheep."
Then Peter, turning about, seeth the disciple whom Jesus loved following; which also leaned on his breast at supper, and said, Lord, which is he that betrayeth thee? Peter seeing him saith to Jesus, Lord, and what shall this man do? Jesus saith unto him, If I will that he tarry till I come, what is that to thee? follow thou me. (Jn 21:20-22)
At the very most, he was a "First among equals" and a frequent "spokesperson" during the early apostolic era.
There is scant evidence that Peter was the one who founded the Church of Rome or even that he spent any significant time to be their "Bishop" but that is neither here nor there. This is of little consequence to me but means EVERYTHING to someone trying to make the case that the Roman Catholic Church is the "One True Church" that Jesus founded in Mt 16.
The bishop of Rome was regarded by other bishops (Patriarchates) as being "First among equals" based upon a traditional understanding that Peter was ultimately the bishop of Rome as well as the fact that the bishop of Rome was closes to the political power and center of the Roman Empire. When the Roman Empire was divided and Constantinople became the political center, the Bishop of Rome rose in prominence to fill this political vacuum and this was pretty much where the modern papacy emerged. The Bishop of Constantinople requested of Pope Gregory that the two would be considered as equals but Gregory refused and insisted upon the primacy of the bishop of Rome which, among other things, led to the great schism of 1054. The primacy of the pope has never been universally accepted by the church and papal infallibility was purely a Roman Catholic invention later on.
These are cold, hard facts. Check your history.
False. Look at what the heads of Constantinople said long before 1054.
“If the Roman See recognizes Pyrrhus to be not only a reprobate but a heretic, it is certainly plain that everyone who anathematizes those who have rejected Pyrrhus also anathematizes the See of Rome, that is, he anathematizes the Catholic Church. I need hardly add that he excommunicates himself also, if indeed he is in communion with the Roman See and the Catholic Church of God ...Let him hasten before all things to satisfy the Roman See, for if it is satisfied, all will agree in calling him pious and orthodox. For he only speaks in vain who thinks he ought to pursuade or entrap persons like myself, and does not satisfy and implore the blessed Pope of the most holy Catholic Church of the Romans, that is, the Apostolic See, which is from the incarnate of the Son of God Himself, and also all the holy synods, accodring to the holy canons and definitions has received universal and surpreme dominion, authority, and power of binding and loosing over all the holy churches of God throughout the whole world. (Maximus, Letter to Peter, in Mansi x, 692). Maximus the Confessor Constantinople 650 Ad
“And why, then, passing by the others, does He converse with Peter on these things? (John 21:15). He was the chosen one of the Apostles, and the mouth of the disciples, and the leader of the choir. On this account, Paul also went up on a time to see him rather than the others (Galatians 1:18). And withal, to show him that he must thenceforward have confidence, as the denial was done away with, He puts into his hands the presidency over the brethren. And He brings not forward the denial, nor reproches him with what had past, but says, 'If you love me, preside over the brethren ...and the third time He gives him the same injunction, showing what a price He sets the presidency over His own sheep. And if one should say, 'How then did James receive the throne of Jerusalem?,' this I would answer that He appointed this man (Peter) teacher, not of that throne, but of the whole world. (Chrysostom, In Joan. Hom. 1xxxviii. n. 1, tom. viii)
John Chrysostom, Patriarch of Constantinople (c. 387)
John VI, Patriarch of Constantinople (715)
The Pope of Rome, the head of the Christian priesthood, whom in Peter, the Lord commanded to confirm his brethren. (John VI, Epist. ad Constantin. Pap. ad. Combefis, Auctuar. Bibl. P.P. Graec.tom. ii. p. 211, seq.)
St. Nicephorus, Patriarch of Constantinople (758-828)
Without whom (the Romans presiding in the seventh Council) a doctrine brought forward in the Church could not, even though confirmed by canonical decrees and by ecclesiastical usuage, ever obtain full approval or currency. For it is they (the Popes of Rome) who have had assigned to them the rule in sacred things, and who have received into their hands the dignity of headship among the Apostles. (Nicephorus, Niceph. Cpl. pro. s. imag. c 25 [Mai N. Bibl. pp. ii. 30]).
St. Theodore the Studite of Constantinople (759-826)
Writing to Pope Leo III:
Since to great Peter Christ our Lord gave the office of Chief Shepherd after entrusting him with the keys of the Kingdom of Heaven, to Peter or his successor must of necessity every novelty in the Catholic Church be referred. [Therefore], save us, oh most divine Head of Heads, Chief Shepherd of the Church of Heaven. (Theodore, Bk. I. Ep. 23)
I did check my history, I quote the patriarchs of Constantinople themselves and it’s greatest scholars long before 1054 and surprise surprise they all refer to the Apostolic See of Peter in Rome as supreme Head of all the Churches.