In the podcasts, Mazza brings up the fact when Christ conferred on Peter what was clearly intended to an office that was to be handed down, when Christ gave Peter the “keys” and clearly made him head of the Apostles, earthly head of the Church built on his “rock”, when Christ at the Sea of Galilee confirmed Peter’s office, intended to be handed down, Peter had not yet been anywhere near Rome.
Peter was Vicar of Christ before he was Bishop of Rome.
When Christ takes the Apostles to Caesarea Philippi (Matthew 16:16-18) he gives people the “keys”. About a year later, the Apostles are ordained at the Last Supper.
Peter has the keys from Matthew 16 onward, but he is not a “bishop” until the Eucharist and Holy Orders were established by Christ.
Consider that when Christ gave primacy to Peter, there were no sees or dioceses. Peter later would found the Church at Alexandria and Antioch. Wouldn’t Antioch have been the primal see? But Peter left Antioch and went to Rome. So there seems to be nothing absolutely necessary about the one we now call “Pope” being Bishop of Rome.
This is a Latin interpretation of the proof texts used to back Rome's claims regarding the papacy. Non-Latins would interpret them differently. Those who hold to Latin ecclesiology are forced into interesting intellectual puzzles...