At first glance, it might seem that the question, of whether Peter went to Rome and died there, is inconsequential. And in a way it is. After all, his being in Rome would not itself prove the existence of the papacy. In fact, it would be a false inference to say he must have been the first pope since he was in Rome and later popes ruled from Rome. With that logic, Paul would have been the first pope, too, since he was an apostle and went to Rome.
On the other hand, if Peter never made it to the capital, he still could have been the first pope, since one of his successors could have been the first holder of that office to settle in Rome. After all, if the papacy exists, it was established by Christ during his lifetime, long before Peter is said to have reached Rome. There must have been a period of some years in which the papacy did not yet have its connection to Rome.