***but it is not when St. Peter was made pope***
Your right, it was right after he was "made Pope."
The gospel does not reflect St. Peter's ascention to the papacy, because it does not directly reflect his taking charge of the bishopric of Rome, although the reference to "Babylon" is his letter indicates that he was in Rome, and of course, his leadership following the Pentecost shows that Peter, as well as all the apostles, were changed men since then.