St. Peter did consecrate the Antiochean bishop, and he did come, serve, and was martyred in Rome, like you said. He also had consecrated a bishop in Armenia. We read of his travels and extensive missionary work in the Acts, so none of this is surprising, or contradictory to the traditional view of him.
It is indeed likely that by the time St. Peter established himself in Rome, there had been a Christian community there.
We have scriptural evidence of St. Peter's primacy among the apostles and we have the evidence of the early Fathers that they considered him the first bishop of Rome.
If there had been a leader of the Roman community prior to St. Peter, it is strange no never have his name mentioned.
By no means do I want to get into an argument about it; I've just been trying to sort it out for myself. There are still some things that don't add up, probably never will.
Like you say, there is ample evidence in the scriptures that he had some sort of special status rather than going into supremacy and all that, but all the apostles were given the same powers as he had, binding and loosing. What Jesus' intent was as to passing them down to generations yet to come, isn't clear in scriptures. My assumption would be that he did intend a structural organization(s) after the apostles died, but scripture is silent about that.
The part I don't get is that James was the presider over the first council in Jerusalem and spoke with authority there, not Peter.
Then there is the mystery of John. I haven't plowed through the scriptures for awhile, but I don't know what his relationship was to Peter and the rest of the church. For all I know, he never died. Now the Spirit spoke to him with some words about the churches [note plural in Asia]. So which is it, church or churches? Who can know at this point? Many of the apostles appeared to exercise their valid ministries totally apart from Peter and outside of his authority. It's just that the Church of Rome ended up prevailing in Western history.
Also, in Revelation we have a model of the New Jerusalem, "14And the wall of the city had twelve foundations, and in them the names of the twelve apostles of the Lamb." There is nothing about a rock or cornerstone or one foundation having eminence over the others there, it sounds equal. It's all shrouded in mystery, but I can't help conjecturing now and then.
Logic tells me that there was a leader of the early Roman community; the fact that the individual isn't named is of no particular significance given that St. Peter's wife wasn't named, a lot of details are missing from that early time. Now Joseph of Arimathea was no apostle insofar as we know, but if the legends are true that he went to England and established a church there, it would have had some kind of authority for a head, but not necessarily a bishop per se, until they much later came under the umbrella of the organized church from Rome. I don't know if that is really true or not; it always kind of intrigued me, especially the legend about the unique hawthorne tree, a clone of which used to be and perhaps still is growing in one of the botanical gardens in London. Sorry I drifted off course with the latter.
And how the Irish figure into all this, I haven't read up on, but there is talk that an early, independent Celtic church existed there.