Doesn’t mention a successor. And why not Antioch: the NT places St. Peter there too?
Well, before the Popes became Pope they were Cardinals or Bishop of a city. For example, before Pope John Paul II was made Pope, he was the bishop of Karcow.
Though, Peter was the first amongst the Apostle. I think in Scripture itself. Peter actions show he had a important role. It is believe in tradition that Peter was probably the oldest of the Apostles. Does Jew tradition say that the elders have seniority? Is it no wonder Jesus pick Peter.
Though Peter himself never stated in the head of all the Apostles, his action speaks for himself.
Consider these verses in the Bible.
Matt. 17:24-25 - the tax collector approaches Peter for Jesus’ tax. Peter is the spokesman for Jesus. He is the Vicar of Christ.
Matt. 17:26-27 - Jesus pays the half-shekel tax with one shekel, for both Jesus and Peter. Peter is Christ’s representative on earth.
Matt. 18:21 - in the presence of the disciples, Peter asks Jesus about the rule of forgiveness. One of many examples where Peter takes a leadership role among the apostles in understanding Jesus’ teachings.
Matt. 19:27 - Peter speaks on behalf of the apostles by telling Jesus that they have left everything to follow Him.
Mark 10:28 - here also, Peter speaks on behalf of the disciples by declaring that they have left everything to follow Him.
Mark 11:21 - Peter speaks on behalf of the disciples in remembering Jesus’ curse on the fig tree.
Mark 14:37 - at Gethsemane, Jesus asks Peter, and no one else, why he was asleep. Peter is accountable to Jesus for his actions on behalf of the apostles because he has been appointed by Jesus as their leader