It wasn't until the First Vatican Council in 1870 that the Roman Catholic Church formally asserted as doctrine that Jesus Christ established the papacy with the apostle Peter and that the Bishop of Rome as Peter's successor bore the supreme authority (primacy) over the whole church.
Obviously, both Eastern Orthodox churches and Protestant denominations deny both of these claims as do I. History indicates that even the concept of papal rule of the whole church was established by slow and painful stages with Leo the first to attempt a biblical and theological bases of the papal claim.
Rome was prominent for the reason that it was the imperial capital with the largest and wealthiest assembly and despite persecutions of all kinds, the Roman congregation was large for its day making it influential.
The reason why early Christian writers, beginning with Irenaeus in the second century, refer to Peter and Paul as founders of the church in Rome and to subsequent bishops as successors of the apostles is because roots were important due to the fact that gnostic teachers were asserting a secret tradition arising from Christ and Gnosticism and orthodox Christianity were for a time locked in a serious struggle for supremacy that only one would survive. Many catholic Christians felt that a list of bishops traced back to Peter and Paul was a sure means of safeguarding the apostolic message. That's the reason why they were created and maintained.
Modern RCC doctrines surrounding the papal office did not yet even exist and no one, church leader or secular author, ever asserted that they did. The Bishop of Rome had honor yes but jurisdiction outside of Rome: no.
Rome's growing influence was a part of the increasingly complex church structure to emerge in the third and fourth centuries. Church organization developed in two important ways: (1) the authority of church councils and (2) the authority of certain bishops over other bishops.
Councils arose when churches in various areas began sending their pastors (or bishops) to meetings to discuss common problems. These were at first irregular, but during the third century these provincial councils began to meet annually. In theory, the bishops from the churches were all equal, but in practice this was seldom the case. The pastors of the churches established by the apostles possessed an informal spiritual prestige, and the bishops from the larger cities exercised authority in certain matters over the pastors from smaller towns. As the church grew it adopted, quite naturally, the structure of the empire.
This meant that the provincial town of the empire became the episcopal town of the church and above the provinces in the empire was the metropolis, so bishops in these larger cities soon supervised the bishops in the provinces of that area. Finally, the empire was divided into several major regions, so within the church, people came to think of the church at Rome exercising authority in Italy, Carthage in North Africa, Alexandria in Egypt, Antioch in Syria, and so on.
As the churches within the province thrust out into the countryside, usually through a preaching tour of the bishop, other churches were established to meet the needs of the converts. At first these churches were cared for by clergy sent out from the city. Ministers who served them, however, were not bishops. They were called "priests" from presbyter, the Greek word for "elder." These priests in the country parishes were consecrated and controlled by the city bishop, but they could administer the sacraments.
Thus, as the fourth century began, the catholic churches were establishing general policies by regular regional councils of bishops and handling day-to-day affairs under the oversight of bishops in each area. General councils of the church arose only after Constantine's conversion. To settle major issues troubling the churches the emperor convened synods of bishops from a wide area. Arles, in 314, was a general council of the churches in the West, and Nicaea, in 325, the first General Council of the whole church. The decrees of these and later councils became the law of the church.
This is how the bishop of Rome soared to major importance on the wings of these developments. The Council of Nicea recognized the bishops of Alexandria, Antioch, and Rome as preeminent in their own areas. Jerusalem was granted an honorary primacy. Thus, by 325 the policy of patriarchates, that is, the administration of church affairs by bishops from three or four major cities, was confirmed by conciliar action.
It was Bishop Damasus (366-384) who first stated "Although the East sent the apostles, yet because of the merit of their martyrdom, Rome has acquired a superior right to claim them as citizens" as the prelude to Leo's grasp for ultimate power of the papacy claiming a dynasty of Peter the Prince of the Church that never existed before.
And for good reason, the Gospels make clear that preeminence among the followers of Christ was not to be according to the pattern of the princes of the world who exercise lordship and authority. Furthermore, Peter continued to be notoriously unstable. Even in the Matthew 16:23 passage Jesus rebuked him and called him "Satan" for not understanding "the things that be of God" and Peter denied his Lord in the moment of crisis. Paul criticized him as an unreliable disciple. Etc... etc... etc... Peter was not the "custodian of the Church." He was, like others, an apostle. But Leo's theory assumes that the grant of authority was not to Peter personally but to his office as bishop of Rome and that identification of authority with a particular office is nowhere in the text.
Leo prevailed because a barbarian attack on Italy made the imperial court at Ravenna desperate for the support of any authority that might help to hold the empire in the West together. Thus, in 445 the Emperor Valentinian issued a decree instructing Aetius, the Roman commander in Gaul, to compel the attendance at the papal court of any bishop who refused to come voluntarily. The emperor's edict turned Leo's claim into law.
The imperial document read: "As the primacy of the Apostolic See is based on the title of the blessed Peter, prince of the episcopal dignity, on the dignity of the city of Rome, and on the decision of the Holy Synod, no illicit steps may be taken against this See to usurp its authority. For the only way to safeguard peace among the churches everywhere is to acknowledge its leadership universally."
If history shows anything, it's that the power hungry always seem to find a way to exploit trouble into opportunity to advance themselves. And Leo
was power hungry and like so many manipulators of troubled times successful too.
Of course, he started a schism in the process that continues to this day.
"Do not be called leaders; for One is your Leader, that is, Christ. "But the greatest among you shall be your servant." -Matthew 23:11
Paul wasn't the one Christ asked to feed the sheep and lambs. Paul didn't call the First Council, nor was it Paul who decided things that day. It was Peter each time. Peter was the custodian of the Church. Paul was the Apostle to the Gentiles.