Why would you assume that? The Keys are described as the keys to the kindgom if heaven. The power to bind and loose is listed separately in Chapter 16. In Chapter 18 the keys are not mentioned, yet the authority to bind and loose is given. While the two are related, there is no warrant from scripture to equate the two.
not the centralized organization, but the body of all believers
I always wondered where in the scripture do people find that strange ecclesiology. The Catholic view is that the boundary of the Church is baptism. It is therefore visible Church; there is not such thing as Church invisible. The apostolic hierarchy of the Church is unseparable from the entire Church. The text you cite in Matthew 18 is clear that the Church, and not the fellowship of believers is the final authority in any dispute.
It can not be proven that Peter was ever in Rome and his epistles, for someone who is supposedly the 1st Pope are somewhat narrow in focus.
It sems at this point beyond doubt that he was (see St. Peter and Rome). Peter's letter establish the apostolic authority with which he and the rest of the apostles speak. This is the fundamental task of the papacy, to preserve a single deposit of faith free from fracture.
if he were the first Pope, why not mention it?
The word "Pope" is of later extraction. His exceptional authority is mentioned in the Keys passage and in several others. It is true however that the power and the role of the papacy grew in post-patristic period as the heresies were combatted. Much of the papal power is derived from the Pope's role as patriarch of the Latin Church, since for most people today the Latin Church is THE Catholic Church. But the Catholic Church as a whole has a far more conciliar structure than the Latin Church.
Just because you say there is no warrant doesn't make it so. Why would I assume that there is? Because it is part of the same thought: I will give you the keys of the kingdom of heaven; whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven." "I tell you the truth, whatever you bind on earth will be bound in heaven, and whatever you loose on earth will be loosed in heaven. --Matthew 16:19; 18:18
The keys are Christ's authority. THey are the authority to bind and loose - and they are something that Jesus gave the whole church (all believers).
2nd, it is NOT strange ecclesiology to believe that the CHURCH is the body of believers. It is Scripture. An organizational structure, be it a centralized hierarchy like the Vatican or the local church is a temporal institution. Christ's CHURCH, His Bride, lives forever. The fellowship IS the Church. Temporal leaders, be they bishops, Pastors, whatever are still subject to the body of believers. Look at what Jesus said in Matt 18 regarding forgiveness. If your brother sin against you, you are supposed to do things in a certain order 1) confront him one on one 2) Bring a witness and if he still doesn't repent 3) bring him before the church - the church being the local body of believers.
The Corinthians dealt with their erring brother not by going through an ecclesiastical order, but by dealing with it as a local church body themselves.
So seeing the Church as separate from the fellowship of believers is a bit of a brainwash. You have been taught this by the ecclesiastical order and you are used to it making sense to you. You're having trouble seeing it any other way. I understand that. To many protestants it makes less sense.
During the initial founding of the church, there were times where there would be councils to decide some point of doctrine. You still had apostles back then too. The church of Christ, however, is His bride. It isn't an institution. It is a living organism.
See these verses below in support that it is the body of believers...
Acts 5:11
Great fear seized the whole church and all who heard about these events.
Acts 8:3
But Saul began to destroy the church. Going from house to house, he dragged off men and women and put them in prison.
Acts 9:31
Then the church throughout Judea, Galilee and Samaria enjoyed a time of peace. It was strengthened; and encouraged by the Holy Spirit, it grew in numbers, living in the fear of the Lord
# Acts 11:26
and when he found him, he brought him to Antioch. So for a whole year Barnabas and Saul met with the church and taught great numbers of people. The disciples were called Christians first at Antioch.
There are many others. But it is clear hear that the Church is not a centralized structure headed by Peter residing in Rome.
As to Peter's being in Rome, I've said he probably was at some point because it is indeed likely that he was martyred by Nero. But where is the Scripture that says such? It doesn't exist. Therefore the claim that it is "beyond doubt" is false. It is not beyond doubt.
Irenaeus is a fair source on your side of the case, but you have to remember he was a disciple of Polycarp who was a disciple of the Apostle John, so Irenaeus is a 3rd generation Christian. He wasn't an eyewitness. But giving him the benefit of the doubt at this point (just for arguments sake, I'm not saying I buy that we have historical proof for Peter in Rome) Irenaeus refers to Peter's missionary efforts in Rome- not his papacy (and yes, I know the term was invented some time later though the Bishop of Rome was active early on- even though his supremacy wasn't recognized until later). So, all we have is that Peter MAY have evangelized the Romans. So far we are batting 0 for 2. Not in Scripture. Not in contemporary/earliest records.
Paul wrote to the church. Does'nt make mention of Peter. Knowing the timing of Roman's writing, it is highly doubtful that he wouldn't have greeted Peter were Peter not the Bishop there. It is argued that Peter wasn't bishop at that particular time. Okay. Whatever. He apparently wasn't at any other time either though he may have been active working with the believers there.
What the Catholic church is asking us to believe is that in Matthew 16, Christ called Peter the Rock upon which he would build his church establishing Peter's primacy above all other apostles. We are then asked to ignore the fact that nowhere else in all of Scripture is Peter referred to as the head of Christ's church on earth. All of the New Testament letters which were written at the time of Peter's ministry fail to mention Peter in the least bit as head of the church. A leader in the church, not disputed. Head, disputed.
Peter, like Mary, is a wonderful Christian for us to study because there are so many things we can learn from them. The vast majority of the doctrinal books, however, were written by Paul. Peter writes to encourage persecuted Christians. Paul writes on meatier areas of doctrine. Peter is not show as head in Jerusalem in Acts 15 (James appears to be). In fact, beyond Pentecost, Peter's role is somewhat tame compared to his role in the gospels.
Back to Acts 15 for a second, you have Peter speaking to the apostles and elders. He stands to speak of how God chose to use him, Peter, to reach the Gentiles. But his is not a keynote. It is after much discussion. He is giving his put on the matter. But he isn't making any decision. Next, Paul and Barnabus share what God did through them and the "whole assembly" was silent listening. Finally, James speaks. When he speaks, he does so as one with authority. It definately appears that James is the chosen leader of this assembly, not Peter. He says "9"It is my judgment, therefore, that we should not make it difficult for the Gentiles who are turning to God..." But even there, It wasn't the Apostles and Elders that made the final decision alone. It was the apostles elders with the whole church who voted on the matter and put the plan into action. Again, while there is organization in the body - the body is not an organization. It is an organism.