The presumption is already there, if Peter gives the keys away. But I don’t think he can. I don’t think Peter had the power to give of the keys.
Because they weren’t really his to begin with.
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Let me first add something to the “Keys” issue. As you said, either the keys were particularly to Peter, or to Peter and others. Since we see no others in Scripture given the Keys, it would be to Peter’s successors. Now, you assert that the successors of Peter are all Christians, or those who speak in accord with Christ’s will.(Forgive me for paraphrasing or misrepresenting). One reason for the Catholic Church’s belief that it was to Peter’s chosen successors (besides the record of historical Bishops of Rome, to whom the Early Church Fathers looked as an authority, a record which is relevant to the topic, but perhaps at another time) is the nature of the “keys”. This passage in Matthew 16 is a clear allusion to another passage in Isaiah 22.
"20. And it shall come to pass in that day, that I will call my servant Eliacim the son of Helcias,
21. And I will clothe him with thy robe, and will strengthen him with thy girdle, and will give thy power into his hand: and he shall be as a father to the inhabitants of Jerusalem, and to the house of Juda.
22. And I will lay the key of the house of David upon his shoulder: and he shall open, and none shall shut: and he shall shut, and none shall open.
23. And I will fasten him as a peg in a sure place, and he shall be for a throne of glory to the house of his father.
24. And they shall hang upon him all the glory of his father’s house, divers kinds of vessels, every little vessel, from the vessels of cups even to every instrument of music.
25. In that day, saith the Lord of hosts, shall the peg be removed, that was fastened in the sure place: and it shall be broken and shall fall: and that which hung thereon, shall perish, because the Lord hath spoken it. "
Here God commands that the existing steward of the king (or regent, Prime Minister…it’s the second in command, the man who speaks with the King’s authority while the King is away) is deposed, and another put in his place. The Bible says that the key of the house of David are placed on him, as a symbol of his authority. It speaks of opening and shutting, as it does in many other instances. Then it talks about “his father’s house”, that is, the line of Eliacim. This position of stewardship is a hereditary one, although prophesied to be a temporary one, since God says the peg will one day be removed.
Compare this with Christ’s statements. Peter is a Rock, on which He will found a Church. Unlike the “peg fastened in a sure place” which will one day fall out to the ruin of all that hangs on it, this Rock will not falter, for the Church built on it will not fall to the powers of Hell.
So it is reasonable, I think, to assume, as you do, that the keys’ authority do not disappear with Peter’s death, but live on, just as Jesus promised to be with us until the end of time. Moreover, it is reasonable, I think, from what we see in Isaiah, to say that this authority passes one to one particular person. In Eliacim’s case, it was a hereditary role. Peter’s office was one of an Apostle. We see in the beginning of Acts that this office was one not meant to be vacant. (As a side note, notice that it is Peter who says this, and in doing so offers an authoritative interpretation of Scripture which seems to have no bearing on the matter in question. I mention this to show that proper authority can interpret Scripture, even if that Scripture in question does not seem to be “blinding proof” of the intepretation.) The assembled pray, draw lots, and Matthias is chosen to take Judas’ place. It is reasonable to assume that the same would apply to Peter’s position after his death. Indeed, it was so, and a successor to Peter, and to his see, was chosen, perhaps by Peter himself, perhaps by the congregation at Rome, or perhaps by other bishops. Either way, he had a successor. Most of the Early Fathers and other sources I’ve read or heard about agree that this was Linus. There is, then, a known successor to Peter, and successors to him, in the person of the Bishop of Rome. Certainly, there are a myriad of texts, documents, and writings from the first century onward which attest to a certain authority held by this man above and beyond that of bishop.
Anyway, that’s just a little bit of my case for papal authority. I’m sure others here can make better ones, and I’d be happy to go into it deeper anytime, but I’ve got to leave for home soon. Feel free to contact me personally, if you’d like.