As seen in the Church Fathers, even if St. Peter is the Rock of Matthew 16, the modern papal doctrines do not follow. Rather, a proper equality of bishops is taught by the Sacred Scriptures and the Church Fathers.
The most discussed passage in the debate over the papacy is Matthew 16:18, and whether or not St. Peter is the the Rock in the passage. The debate often boils down to discussing whether or not the Rock is St. Peter, his confession, or Jesus Himself. There are many reasons to think that the passage is not about St. Peter, but about Jesus and his relationship to Jesus. But what if, for sake of argument, we agree with the Roman Catholic apologist? What if we say that St. Peter is uniquely and specifically the rock? This still does not lead to a specific, Petrine, Roman office with universal jurisdiction and infallibility. There was a theme in the early Church writings of St. Peter being the origin for the episcopate itself. So any responsibility or privilege given to the St. Peter was then given to the rest of the episcopate. He may have been uniquely and personally the Rock (though this isn’t explicit in all of the relevant passages), but any uniqueness in office or authority was the foundation and locus of power for the entirety of the Church, not just the bishop of Rome.
This idea of St. Peter being the foundation for all bishops and not just the Roman bishop is very clear in several of St. Cyprian of Carthage’s writings. In St. Cyprian’s Treatise One, he says the following,
And although to all the apostles, after His resurrection, He gives an equal power, and says . . . [John 20:21] . . . yet, that He might set forth unity, He arranged by His authority the origin of that unity, as beginning from one. Assuredly the rest of the apostles were also the same as was Peter, endowed with a like partnership both of honour and power; but the beginning proceeds from unity.
St. Cyprian: Treatise One: Paragraph Four
St. Cyprian acknowledges that all of the apostles had equal authority, but they all are united around the person of St. Peter. While St. Peter may uniquely be the rock and uniquely have been given the keys in Matthew 16, all of the apostles are endowed with his privileges, as they ‘were also the same as was Peter, endowed with a like partnership of both honour and of power...’
St. Cyprian reiterates this train of thought in his Epistle 26, where he says this,
Our Lord, whose precepts and admonitions we ought to observe, describing the honour of a bishop and the order of His Church, speaks in the Gospel, and says to Peter: Thence, 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.
St. Cyprian: Epistle 26: Paragraph One
After quoting the relevant passage in Matthew, he ties it directly to the work of all the bishops. And as seen in the earlier passage, this is centred around the person of St. Peter. Every bishop can trace his origin to St. Peter. There is not singular or unique bishop who is descended from St. Peter: every bishop is. St. Cyprian certainly did not think that St. Peter's episcopal successor in the city of Rome specifically had St. Peter's authority, because to him, every bishop had St. Peter’s authority.
St. Augustine viewed St. Peter as symbol of the entire Church. In his 50th tractate on the gospel of John, he says,
For if in Peter's case there were no sacramental symbol of the Church, the Lord would not have said to him, . . . [Matthew 16:19] . . . If this was said only to Peter, it gives no ground of action to the Church. But if such is the case also in the Church, that what is bound on earth is bound in heaven, and what is loosed on earth is loosed in heaven...
St. Augustine: Tractate 50 on John: Paragraph 12
St. Peter is a unifying figure for the Church. He was the leader of the apostles. This is an authentically catholic view of St. Peter. But this does not in any way entail an office with universal jurisdiction, the charism of infallibility, or that is specifically centred in Rome. St. Peter is not an agent who has these powers of binding and loosing. He is the representative of the whole Church, and as such, the whole Church (which is built on the bishops according to St. Cyprian) has the powers of the keys of the kingdom of heaven.
St. Augustine continues along this line of thought in his 124th tractate on John,
For petra (rock) is not derived from Peter, but Peter from petra; just as Christ is not called so from the Christian, but the Christian from Christ. For on this very account the Lord said, 'On this rock will I build my Church,' because Peter had said, 'You are the Christ, the Son of the living God.' On this rock, therefore, He said, which you have confessed, I will build my Church. For the Rock (Petra) was Christ; (cf. 1 Corinthians 10:4) and on this foundation was Peter himself also built. For other foundation can no man lay than that is laid, which is Christ Jesus. (cf. 1 Corinthians 3:11) The Church, therefore, which is founded in Christ received from Him the keys of the kingdom of heaven in the person of Peter, that is to say, the power of binding and loosing sins. For what the Church is essentially in Christ, such representatively is Peter in the rock (petra); and in this representation Christ is to be understood as the Rock, Peter as the Church.
St. Augustine: Tractate 124 on John: Paragraph 5
In this particular articulation of St. Augustine, St. Peter is not even the Rock. He was a symbol that represented the Catholic Church, which received the keys in the person of St. Peter. The keys were never something that belonged solely to St. Peter.
This is reiterated a third time in St. Augustine’s sermon on Sts. Peter and Paul. He says,
It was in the person of the whole Church, which he alone represented, that he was privileged to hear, To you will I give the keys of the kingdom of heaven. After all, it is not just one man that received these keys, but the Church in its unity. So this is the reason for Peter's acknowledged pre-eminence, that he stood for the Church's universality and unity, when he was told, To you I am entrusting, what has in fact been entrusted to all.
St. Augustine Sermon on Sts. Peter and Paul
St. Augustine makes it extremely clear that St. Peter does not act in his own person. He is a symbol for the Church. St. Peter receiving the keys was the Church receiving the keys. There is no one singular line of episcopal succession that is vested with St. Peter's authority. The entire Church is vested with St. Peter's authority.
St. Optate in his book against the Donatists, explicitly conflates the chairs of Sts. Peter and Cyprian. He writes,
nor was it Caecilian who separated himself from the Chair of Peter, or from the Chair of Cyprian ----but Majorinus,
St. Optate: Against the Donatists: Book One Chapter 10
To St. Optate, to separate from the Chair of St. Cyprian in Carthage was to separate yourself from the Chair of St. Peter.
St. Peter being in the person of the Church is also stated in Tertullian's work Scorpiace. He says,
For though you think heaven still shut, remember that the Lord left here to Peter and through him to the Church, the keys of it, which every one who has been here put to the question, and also made confession, will carry with him.
Tertullian: Scorpiace Chapter 10
Once again, St. Peter is seen as a representative of the Church, not its sole, infallible head. The keys are available to those who make St. Peter's confession.
Pulling the Threads Together
The idea that St. Peter had unique authority over the other apostles and that he and his successors specifically in Rome had universal jurisdiction and infallibility is foreign to the first several centuries of Church history. While it may be easy to see these modern papal doctrines in the Fathers if one already believes and presupposes them, they cannot properly be derived from the Fathers. Contrary to the modern papal doctrines, St. Peter's seemingly unique authority was translated to the rest of the apostles, and consequently, the rest of the bishops. As St. Cyprian pointed out, Matthew 16 applied to all of the bishops, who were all rooted in St. Peter's episcopate. St. Peter is not a separate person who is in the genus of the episcopate who was higher than the others. It is authentically Catholic (and therefore Anglican) to say that St. Peter was a head, leader, and mouthpiece for the apostles. But as St. Cyprian says, all the other apostles were equal in honour and power. This is the Catholic view of St. Peter. He was a vocal leader and a key figure in the establishing of the early Church, through whom all the other apostles received their authority, from Jesus Christ. Roman Catholics try to prove too much with what we have from the early Church. There is no reason (and many reasons to the contrary) to think that St. Peter and his specifically Roman successors had any authority over the other apostles or subsequent bishops.
To the extent that St. Peter was seen as the Rock of the Church in his person, it was then universally communicated to the episcopate. So even when St. Peter is uniquely, personally the Rock, he isn’t! This along with the fact that a lot of fathers also saw the Rock in Matthew 16:18 as St. Peter’s confession or even Christ Himself, goes to show that the modern and necessary Roman Catholic reading of Matthew 16:18 is not historical, contrary to the claims of Vatican Council I.