Why does St. John, who lived until around A.D. 100, not mention in his canonical writings the popes who immediately followed Peter, such as Linus, Anacletus, and Clement I? This seems odd, given that these men functioned as the Vicar of Christ, the visible head of the Church.
This is a very rough analogy, and it must be remembered that John, as an original apostle, had universal jurisdiction although in subordinance to Peter’s primacy; but look at it this way:
Let’s say you are governor of a state. Two thousand years from now, historians look back at the missives you wrote to address problems in your state. Because you do not happen to mention any U.S. presidents of the time, should the historians assume that there were none?
Silence on the matter of popes succeeding Peter does not prove one way or the other that there weren’t any. It only demonstrates that the apostle John had other things on his mind at the time of drafting these letters and simply did not mention them. Unless there was some pressing reason that he do so – such as a dispute over their authority – it makes sense that he would focus on the topics he meant to discuss and not digress into passages like: “After Peter there were Linus, Anacletus, and Clement; but you already know this, don’t you?”