We have proven that Mat. 16:18 does not mean what modern Rome claims it means. But we must examine the consequences they take from their novel interpretation so let's for the sake of argument treat the passage as meaning some special leadership charisma for Peter and Peter exclusevily.
And let's deal with (2) - even it meant what modern Rome claims it to mean, would it have the consequences they claim it has?
What consequences are that? Namely,
(a) Necessary, untransferable leadership of Peter and his successors over the Church, meaning the bishops of Rome;
(b) this leadership is necessarily exercized through universal jurisdiction, meaning that no bishop is autonomous in relation to the leader of the Church, who can even override bishops' and councils when necessary;
(c) Infallibility in questions of dogma and morality;
Let's examine (a).
At once we meet one problem: the Patriarch of Antioch is also a successor of Peter and it is reasonable to say that the Patriarch of Alexandria, who is the successor of Mark, disciple of Peter is also a successor of Peter. But if we keep just the Patriarch of Antioch we immediatelly have the situation that, if any special charisma was given to Peter *and* his successors, everything that is said of the bishop of Rome is also true of, at least, the bishop of Antioch. This is a very underestimated fact because it is a final proof that even if Romans were right about the principle they would necessarily be wrong about the consequences. Each line of successors of Peter would have inherited the "special charisma" of primacy and we would have at least a diarchy or even a triarchy in the Church. It couldn't mean the bishops of Rome only.
On what basis does modern Rome claim, then, that this charisma belongs *only* to the successors of Peter *in Rome*?
Mostly on an ex post facto interpretation of letters among church illustrous figures, the titles assigned to the pope, and requests of intervetion from other sees to Rome. Also on the colorful events like the proclamation of Chalcedonian bishops that "Peter spoke through Leo".
Regarding the titles, we don't have to look any further than the title of that other successor of Peter who is the Patriarch of Alexandria, styled "Judge of the World". To modern ears it strikes as an utter presumption, but for Greek-Roman style no hyperbole was too much to pamper an authority. That hyperbolic titles were banal is easily verifiable in any analysis of texts of the first period, specially in Greek. Even the librarian of the imperial palace was styled "Librarian of the Universe" and there were several "protos" and "archi", that it, "first" this, "principal" that. Even today we have "archdeacons", "archpriests" and "archbishops". Despite the fact we have thousands of them, no one would take the prefix "arch" (principal, main, chief) literally, meaning that the "archdeacon" is in fact the universal leader of all deacons. I believe that as the West lost contact with Hellenistic culture and greek, these stylistic matters became transparent to them and titles that were long used were taken to their literal meaning - the tendency to "objectivity" and the search for the "clarity" that is typical of the West ironically blinding us to the subtleties of a culture that was already abusing these figures of speech, but with the waiver that, being literate in their own mother-language, they knew these were hyperboles and not literal descriptions.
For the requests of intervention, what is not said is as important as what has been said, that is, churches asked the moderation of each other often, not only Rome. Even when Rome was requested to moderate, we don't see any trace of "Rome spoke, the case is closed" motto, for the litigation would continue as long as both parties were strong enough to resist the other. One thing that did set Rome apart from the other Churchs is that it had centuries of history on the side of the Orthodox Faith, not being the source or nest of any large or serious heresy and that was always remembered. It is clear that Rome was held in high steem because it had kept the Orthodox Faith. Only later, papism would claim an inverted order: that the Orthodox Faith had been kept because of the high position it had, as a provision of God for men. Every single text praising Rome, in its proper dialogical context, that is, inserted in the conversation it was part of, can be shown to argue precisely that "Rome is worthy of praise and dignity because it has always been Orthodox" and not once we can see "Rome will always be Orthodox because God put her in a special place of dignity". Nowhere it is implied "automatic Orthodoxy" for Rome.
Finally, in the event of Chalcedon, it is known that a common interpretation of the time about the bishops is that *each one of them* was a successor of Peter, and the Throne of Peter was the collegiate of all bishops, just like Jesus refers to all the Pharisees as sitting on the Throne of Moses. "The throne" was a role, that could and was played by all the authorities of the people of God, and the change from the image of this authority from "throne of Moses" to "throne of Peter" adds to the point that Peter was seen as a transitory figure of authority like Moses, who was the leader of the Hebrews in their transition to the Promised Land, but instituted a collegiate of Judges, just like after the Apostles we had the collegiate of bishops. The special ministry was necessary a bit longer with Joshua but soon after it became a governance by confederate judges, just like the church was to be led by confederate bishops. That this parallel to the original judges was pretty much clear for the early Church we know from the roles the bishops had in being actual judges in the Church, and St. Paul even ask the people to not use the secular (Roman) tribunals, but to settle everything internally in the church, with the "new judges", the bishops. The parallels between Moses and Peter are many and would demand a study of its own. Enough to say that Peter led different peoples to the "promised land" of the Kingdom of Christianity, being the first "Converter" for Jewish, Romans and other pagans as seen in Acts, leading them away from the "Egypt" of their passions. Moses is indeed a prefiguration of Peter, just like the Judges are of the bishops, and the monarchical troubles and eventual fall of the Hebrews a warning for the future of the Western Church.
What this all means related to Chalcedon is that the "Voice of Peter" that spoke through Leo, was the collegiate of bishops gathered in the synod itself. What they were saying in "plain English" was that Leo had voiced the common agreement of all bishops, and nothing more.
So, as seen above, the praxis of the 1st millenium reflects a collegiate governance, with preeminance given due to a history of "experience" in "being Orthodox". It does not assume or postulates a see miraculously unable of falling into heresy, an posterior assumption that can only be projected onto those facts, but not concluded from them.
With this we have proven that (a) was not and could not be a consequence of Mat 16:18 even if it meant what modern Rome claims it to mean.