Antioch, and a lot of the other cities first colonized by the apostles, were a lot more wide open than Rome. Rome was where imperial politics happened. Rome had extremely good secret police forces that were supposed to get rid of traitors.
The emperors, from Augustus on, were all promoting their emperorship as something divinely mandated by the gods of all nations, with the emperor as a god (or at least, his guardian spirit or “genius”) and descended from the Olympian gods (Julius Caesar claimed descent from Venus). Also, there was that whole messy Jewish messianic revolt thing.
From the Roman point of view, Christians advocated the rule of the world by an immortal god emperor, who would overthrow all other earthly kings and especially the Romans (as seen in the Book of Daniel). Having a few Christians living in the Jewish community, or among the general population, was one thing. Having a Christian bishop there… well, that was entirely another.
Any bishop who went to Rome to lead the Christian community there, was begging to be arrested, convicted, and executed for treason. So until the entire Church was really geared up for everything that had to be done there, it would have been stupid to throw one of the apostles with bishop powers to the lions.
The traditional story (told by St. Justin Martyr and other sources) is that Peter went to Rome because Simon Magus (the simony guy from Acts) was running around misrepresenting Church doctrine and working false miracles, pretending to be Christ or a god. Peter had to go there to fight heresy and save souls. After publicly defeating Simon Magus in a big way, Peter found a lot of ex-Magus followers demanding to become Christians, so from then on, he was too busy with the Church in Rome to ever move back to Antioch on a permanent basis.
And honestly, I don’t see any reason not to believe that. It was a very big step, and it meant travelling far to the west, out of the Semitic and Greek world of the early Christians’ comfort zone. You also needed a lot more personnel to administer Rome – tons of deacons just for single neighborhoods – and there was a lot more poor people to take care of. It was a huge venture for the Church; and in human terms, it all blew up on them in Nero’s persecution, when huge numbers of the new Christian community were imprisoned and then killed. There’s no good reason that Peter would have gone to Rome, unless he was under orders from God (or emergency necessity of fighting heresy) to bite off more than he could chew.