First the material that inspired this post.
From the opening statement of Eastern Orthodox Chris in a debate with Catholic apologist Scott Windsor:
What is Primacy?
At an Eastern Church service where multiple clergy are present, every bishop and priest has a position of rank, and each takes his position in the service based on that. This doesn’t mean that the higher ranks lord it over the lower ranks. It is an honorary status, a right to chair a meeting if you are the highest rank present. Some quotes will shed light on the early church’s understanding of Peter’s primacy:
Peter and John were equal in dignity and honour. Christ is the foundation of all - the unshakable rock upon which we are all built as a spiritual edifice. ~Cyril of Alexandria to Nestorius.
He has not the primacy over the disciples but among the disciples. His primacy among the disciples was the same as that of Stephen among the deacons. ~Augustine, Sermon 10 on Peter and Paul.
But observe how Peter does everything with common consent; nothing imperiiously. ~John Chrysostom, Homily III on Acts 1:12
To all the apostles after His resurrection He gives equal power (parem potestatem) and says "As the Father hath sent me, even so I send you: " ~Cyprian, De Unitate 4.
For neither did Peter, whom first the Lord chose… when Paul disputed with him afterwards about the circumcision, claim anything to himself unsolently, nor arrogantly assume anything, so as to say that he held a primacy, and that he ought to be obeyed by novices and those lately come. Cyprian, Epistle LXX concerning the baptism of heretics.
In the administration of the Church each bishop has the free discretion of his own will, having to account only to the Lord for his actions. None of us may set himself up as bishop of bishops., nor compel his brothers to obey him; every bishop of the Church has full liberty and complete power; as he cannot be judged by another, neither can he judge another (Cyprian’s opening address to the Council of Carthage.
… through the changes of times and successions, the ordering of bishops and the plan of the Church flow onwards; so that the church is founded upon the bishops and every act of the Church is controlled by these same rulers. Since this then is founded on the divine law, I marvel that some, with daring temerity, have chosen to write to me as if they wrote in the name of the Church. ~Cyprian to the Lapsed, Epistle XXVI.
Where were the cries of heresy to these statements of Cyprian and others if the Bishop of Rome were the supreme head of the Church?
Peter and the Rock
A large part of the Catholic claim about the position of the Pope comes from the claim that Peter is the rock upon which Christ builds his Church as per Matthew 16:18. … [S]ome Catholic apologists will say that we can’t distinguish between Peter’s confession and Peter himself. This makes no sense at all, because his confession is one that every Christian confesses, it is not unique to Peter. But in any case, it is the Fathers themselves who clearly distinguish between Peter and his confession, so we need not speculate on this:
Faith is the foundation of the Church, for it was NOT OF THE PERSON but of the faith of St Peter that is was said that the gates of hell should not prevail against it, it is the confession of faith that has vanquished hell. Jesus Christ is the Rock. He did not deny the grace of His name when he called him Peter, because he borrowed from the rock the constancy and solidity of his faith. Endeavor then thyself to be a rock - thy rock is thy faith and faith is the foundation of the Church. If thou art a rock, thou shall be in the Church, for the Church is build upon the rock" ~St Ambrose, On the Incarnation.
“Rock is the unity of faith, not the person of Peter.” ~St Cyprian, De Catholicae Ecclesiae Unitate, cap. 4-5
More to the point, of all the Fathers who interpret these passages in the Gospels (Matt 16:18, John 21:17), not a single one applies them to the Roman bishops as Peter’s successors.