The only way Peter could have founded the church at Rome was if he had appointed/commissioned the first leader of the church and sent him there. Paul wrote to the Romans and makes no mention of Peter's being there which is odd since he names so many others.
By the time Peter finally got to Rome, which I believe he did, and wrote a letter from there with the code word "Babylon" (I don't think he was literally at the ancient Babylon), someone else had already been leading the church; it could have been informally organized by early lay people, like I believe many others probably were in the earliest times, the leaders may have been "ordained" by the laying on of hands by one of the apostles.
That is not to say he couldn't have been the first pope, but I'm not fully convinced he was. It doesn't take many years for historians to get wrong information recorded that was handed down by word of mouth, and sadly, I think some writings were later altered to give more substance to claims.
I've read all the claims; none are totaly convincing one way or another.
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.