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How can he just give over his understanding of Scripture to the authority of Rome?
How? Here's how - because Jesus told Peter, the first Pope,

"And I say also unto thee, That thou art Peter, and upon this rock I will build my church; and the gates of hell shall not prevail against it. And I will give unto thee the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.

The overwhelming majority of the Church Fathers agreed that Matthew 16:18 that you cited applied to every genuine Christian, not to Peter exclusively.

Augustine sums it up nicely:

But whom say ye that I am? Peter answered, ‘Thou art the Christ, The Son of the living God.’ One for many gave the answer, Unity in many. Then said the Lord to him, ‘Blessed art thou, Simon Barjonas: for flesh and blood hath not revealed it unto thee, but My Father which is in heaven.’ Then He added, ‘and I say unto thee.’ As if He had said, ‘Because thou hast said unto Me, “Thou art the Christ the Son of the living God;” I also say unto thee, “Thou art Peter.” ’ For before he was called Simon. Now this name of Peter was given him by the Lord, and in a figure, that he should signify the Church. For seeing that Christ is the rock (Petra), Peter is the Christian people. For the rock (Petra) is the original name. Therefore Peter is so called from the rock; not the rock from Peter; as Christ is not called Christ from the Christian, but the Christian from Christ. ‘Therefore,’ he saith, ‘Thou art Peter; and upon this Rock’ which Thou hast confessed, upon this rock which Thou hast acknowledged, saying, ‘Thou art the Christ, the Son of the living God, will I build My Church;’ that is upon Myself, the Son of the living God, ‘will I build My Church.’ I will build thee upon Myself, not Myself upon Thee. For men who wished to be built upon men, said, ‘I am of Paul; and I of Apollos; and I of Cephas,’ who is Peter. But others who did not wish to built upon Peter, but upon the Rock, said, ‘But I am of Christ.’ And when the Apostle Paul ascertained that he was chosen, and Christ despised, he said, ‘Is Christ divided? was Paul crucified for you? or were ye baptized in the name of Paul?’ And, as not in the name of Paul, so neither in the name of Peter; but in the name of Christ: that Peter might be built upon the Rock, not the Rock upon Peter. This same Peter therefore who had been by the Rock pronounced ‘blessed,’ bearing the figure of the Church ---Philip Schaff, Nicene and Post-Nicene Fathers (Grand Rapids: Eerdmans, 1956), Volume VI, St. Augustin, Sermon XXVI.1-4, pp. 340-341

In the Matthew 16 passage, Peter is a figurative representative of the church.

Augustine further makes the expressed point that all the Apostles were equals in all aspects, and that each receives the keys, as with all genuine Christians, and not Peter exclusively.

Before his passion the Lord Jesus, as you know, chose those disciples of his, whom he called apostles. Among these it was only Peter who almost everywhere was given the privilege of representing the whole Church. 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’ (Mt 16:19). After all, it isn’t 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.

I mean, to show you that it is the Church which has received the keys of the kingdom of heaven, listen to what the Lord says in another place to all his apostles: ‘Receive the Holy Spirit;’ and straightway, ‘Whose sins you forgive, they will be forgiven them; whose sins you retain, they will be retained’ (Jn 20:22-23). This refers to the keys, about which it is said, ‘whatever you loose on earth shall be loosed in heaven, and whatever you bind on earth shall be bound in heaven’ (Mt 16:19). But that was said to Peter. To show you that Peter at that time stood for the universal Church, listen to what is said to him, what is said to all the faithful, the saints: ‘If your brother sins against you, correct him between you and himself alone’---John Rotelle, Ed., The Works of Saint Augustine (Hyde Park: New City, 1994), Sermons, III/8 (273-305A), On the Saints, Sermon 295.1-3, pp. 197-198

Augustine, along with the vast majority(95%) of the church fathers did not interpret any Roman primacy through Peter in the slightest, quite the opposite actually.

Augustine's very protestant teaching on Matthew 16:18:

Christ, you see, built his Church not on a man but on Peter’s confession. What is Peter’s confession? ‘You are the Christ, the Son of the living God.’ There’s the rock for you, there’s the foundation, there’s where the Church has been built, which the gates of the underworld cannot conquer.---Sermon 229