Any reference to a key is an allusion to what keys are used for (opening things kept hidden). In Peter's case, the usual understanding is that he was chosen by Christ to open the world to the Gospel. Remember that this was not the case before Pentecost. There is no necessary connection between the keys and the power to bind and loose, but IF THERE WERE, it would apply to Peter, not to Popes.
The main issue with the Papacy is the claim of men who were NOT the recipients of any promises to have appropriated them for themselves merely by being the bishops of Rome.
In your effort to refute the Papacy you are ignoring the connection to Isaias 22.
You are saying that the keys are an allusion to opening and closing and you are correct butnot in the way you see it.
Consider what Isaias 22:22 is saying.
Isaias 22:22. 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.
As you said the Key opens and closes. But it is not things kept hidden. Jesus gave the world Truth and Grace and he was not hiding it.
The Key(s) has a very real meaning and one that commissions someone to an office.
As in Isaias 22:22 above we see that the Key of the House of David gave him the power to open and close and no one could change it. This is not speaking of the doors in the house but of laws and decisions.
In Matthew 16:19 the same things is said almost word for word when Peter (alone) receives the Keys.
Matthew 16:19. And I will give to thee the keys of the kingdom of heaven. And whatsoever thou shalt bind upon earth, it shall be bound also in heaven: and whatsoever thou shalt loose on earth, it shall be loosed also in heaven.
Here Jesus gave Peter the Keys and says whatever he binds or looses that no man can change. But what is different in Matthew compared to Isaias is that we have heaven and earth and not just the House of David.
Binding and Loosing are the same as opening and closing from Isaias 22:22.
To open is to loose and to close is to bind.
But consider too Matthew 16:18 because Matthew 16:19 are speaking of the Church. So we can see that Jesus' Church will be in heaven and on the Earth and these Keys give Peter the power to bind and loose for the Church. And it must be this Church that Jesus speaks of.
Matthew 16:18. And I say to thee: That thou art Peter; and upon this rock I will build my church, and the gates of hell shall not prevail against it.
Matthew 16:19 is a continuation of Matthew 16:18 because it starts with AND.
So the Keys are cannot be an allusion to anything but Jesus' Church and Peter having a very specific power to bind and loose. This power is the same as in Isaias since that is what the Key represents.
When you come to the realization that Matthew 16:19 is a direct relation to Isaias 22:22 then you will know it is an office that is filled by another when one leaves his office.
Isaias 22:19. And I will drive thee out from thy station, and depose thee from thy ministry.
"depose thee" from his station... obviously it is a station (or office) that once vacant is filled again.
Isaias 22:20. And it shall come to pass in that day, that I will call my servant Eliacim the son of Helcias
Eliacim is filling a now empty office. It is this way with the Pope as well. That is why that has been a well maintained history of the seat of Peter since Peter.
Keep in mind that this office of Peter also bears a direct connection to the power of the Seat of Moses. And what is the Seat of Moses? Scripture mentions it but once when Jesus says to do as the Pharisees teach but not as they do for they sit in the seat of Moses.
Matthew 23:1. Then Jesus spoke to the multitudes and to his disciples,
23:2. Saying: The scribes and the Pharisees have sitten on the
chair of Moses.
23:3. All things therefore whatsoever they shall say to you, observe and do: but according to their works do ye not. For they say, and do not.
23:4. For they
bind heavy and insupportable burdens and lay them on men's shoulders
Notice too the use of the word "bind" as mentioned here.
The understanding of the KEYS has to be in accord with all scripture and the Catholic teaching of this is in accord with all scripture with it's Traditions both of the Old and the New Testaments.