Pope Impossible

The Apostles of Jesus Christ were scattered far and wide. How reliable was communication between the Christians of India, Ethiopia, Turkey and Spain? Is it unrealistic to claim Peter’s authority over all these lands in a time when it would have been a matter of practical impossibility? Early church fathers did not interpret Matthew 16:19 as Rome later came to. Could it simply be that wherever the right faith is to be found, there are the keys to the kingdom? How can any one person have a monopoly on that faith? Is that why Paul wrote that no other foundation than Jesus Christ can be laid (1 Corinthians 3:11)?