Balderdash. Flat out wrong. These traditions were already firmly in place--and more importantly not questioned--by the second century, more than 100 years before Constantinople was even founded.
Ignatius wrote to the Roman Church in 110: "Not as Peter and Paul did, do I command you". Dionysius of Corinth in 170 says flat out that Peter and Paul founded the Church of Rome. Irenaeus relates it again in the 160s-170s. Gaius mentions the Tropaion of the Apostles at the Vatican in around 200--can that refer to anyone else but Peter? This tropaion, as part of a "red wall complex" under the Vatican, was excavated a few decades ago and the bricks in it dating to the Caesarship of Marcus Aurelius between 147-161.
As for the "Babylon" connection, let's look at what the Assyrian Church of the East has to say--because unlike Germanic johnny-come-latelies to this issue, these are the folks who are lineal descendants from the original Christians in Babylon. There is actually a belief among them that the Petrine epistle was in fact written from their homeland in Babylon proper. And yet, whaddya know...the Syriac liturgy in multiple places nevertheless asserts that Peter and Paul founded the Church in Rome. If anyone had a reason to refute the Peter-Rome connection, they did. And yet they did no such thing--even after 1500 years of schism with the Church of Rome.
I'll return to your other larger points if and when I have a chance.
Utter nonsense. Irenaeus' imagination notwithstanding, the fact remains that the matter of Roman supremacy was not firmly settled until the papcy of Leo the Great (440-461 AD).
Ignatius wrote to the Roman Church in 110: "Not as Peter and Paul did, do I command you". Dionysius of Corinth in 170 says flat out that Peter and Paul founded the Church of Rome. Irenaeus relates it again in the 160s-170s.
It's entirely unclear that Ignatius was suggesting that Peter had been in Rome preaching. Indeed, the passage you reference begins with "I write to all the Churches, and declare to all men, that I willingly die for the sake of God, if so be that ye hinder me not....", followed by Ignatius' request that the Romans pray for him to have constancy in martyrdom, and then the statement that you cite. However, Ignatius is not saying that Peter and Paul had both been in Rome, commanding them. Rather, he simply and only references the fact of their apostolic authority, contrasting it with his own position as one condemned. Contextually, there's nothing here that suggests what you're saying. At best, it could imply that - as was the case with Ignatius himself - Peter was brought to Rome for martyrdom (though that is also by no means settled on evidentiary grounds). The reference is about authority, not geography. Indeed, from what Ignatius writes, it would seem that Peter and Paul were also commanding the "other churches" that Ignatius alludes to - which again suggests their apostolic authority is in view, likely due to both their positions as "primary apostles" and that they were the two major vessels used by the Holy Spirit to write the New Testament.
As for Dionysius, the problem with the fragments we have left is that they're not a trustworthy source. The last fragment includes a complaint that his own letters were being tampered with by "apostles of the devil" who would add to and take away. This puts us into sort of a logical quandary, since how do we know that anything in them (including, ironically, the complaint itself) is genuine? Nevertheless, shaky evidence which may very well have been adulterated to include fabricated support for the growing move to attribute a primacy to Rome.
As for Irenaeus, well, what can we say about him? True, he does on several occasions mention that Peter and Paul "laid the foundations" of the Roman church, etc. But that's just the problem - it's scripturally impossible. Paul's epistle to the Romans is dated at around 57-58 AD. We know from the chronology of Acts that Paul had not been to Rome prior to this date. When Paul writes to the Romans, he indicates that he had not seen them, but was writing to them out of reputation (similar to what we see with the Colossians church) - Romans 1:10,15. Paul writes to this as-yet unmet group of Christians who were already assembled as a church. Without him. Not started by him, nor even built up yet by him. Further, Catholic tradition says that Peter was pope in Rome between 41-66 AD. Are we to presume that Paul would write an epistle to the church where the First Pope of all Christianity was pastor, and not greet this fellow apostle, or even mention his presence? Ridiculous. Clearly, Irenaeus is wrong on this count. This being so, it seems that Irenaeus is writing what he did more for the purpose of bolstering some claim on Rome's part to special apostolic position - and this could very well apply to Peter as well. Irenaeus is not a trustworthy witness in this regard, and in fact, has his problems throughout. He is, after all, the same guy who thought Jesus died when he was 50, despite the obvious Scriptural impossibility of this assertion.
Gaius mentions the Tropaion of the Apostles at the Vatican in around 200--can that refer to anyone else but Peter? This tropaion, as part of a "red wall complex" under the Vatican, was excavated a few decades ago and the bricks in it dating to the Caesarship of Marcus Aurelius between 147-161.
Of course it can refer to somebody else besides Peter - there were a dozen other apostles besides him, after all. Further, Gaius says that the trophies of the Apostles are at both the Vatican and the Ostian Road. Later tradition associates these with Peter and Paul, but this is a century and a half after the fact. Gaius may be relating a developing tradition (or, likewise, he may have had not the least idea of Peter or Paul specifically in his mind when he wrote this - we'll never know since his works exist only as fragments, which limits our ability to delve context), but as with Irenaeus, the problem with the tradition is that it doesn't accord with Scriptural facts.
As for the "Babylon" connection, let's look at what the Assyrian Church of the East has to say--because unlike Germanic johnny-come-latelies to this issue, these are the folks who are lineal descendants from the original Christians in Babylon. There is actually a belief among them that the Petrine epistle was in fact written from their homeland in Babylon proper. And yet, whaddya know...the Syriac liturgy in multiple places nevertheless asserts that Peter and Paul founded the Church in Rome. If anyone had a reason to refute the Peter-Rome connection, they did. And yet they did no such thing--even after 1500 years of schism with the Church of Rome.
Once again, there are Scriptural deficiencies with this assertion of tradition. Tradition cannot overturn Scripture, so the Syriac traditions and liturgies are wrong. Further, archaeology would seem to agree with Scripture. There's a solid case to be made that Peter's ossuary was discovered in Jerusalem in 1953. Also, what are we to make of the fact that when Venerando Correnti, an Italian anthropologist, examined in 1956 the bones that had been certified by Pius XII as belonging to Peter, he found out that they were really an assortment of bones from a number of individuals, some of them female, and some not even homo sapiens. These were the bones of the "Red Wall", that had been certified as "Peter's Tomb." Clearly, these were not the bones of Peter, despite Pius XII's wishful thinking.