Would you do me the great favor and tell me the names of one or two of the early Christian writers who name Peter as Pope or Bishop of Rome?
That would save me the trouble of hunting through everyone to convince myself of a negative, and then I would find that I had missed one.
St. Clement, a disciple of the Apostles, who, after Linus and Anacletus, succeeded St. Peter as the fourth in the list of popes. In his "Epistle to the Corinthians", written in A.D. 95 or 96, he bids them receive back the bishops whom a turbulent faction among them had expelled. "If any man", he says, "should be disobedient unto the words spoken by God through us, let them understand that they will entangle themselves in no slight transgression and danger" (Ep. 59). Moreover, he bids them "render obedience unto the things written by us through the Holy Spirit".
St. Ignatius of Antioch, ca. 107 A.D., in the opening of his letter to the Roman Church, refers to its presiding over all other Churches.
St. Irenaeus, a disciple of St. Polycarp, who had been appointed Bishop of Smyrna by St. John, in "Adversus Haereses" (3:3:2), cites the Apostolic tradition faithfully preserved by the Church from the twelve Apostles. He notes that the See of Rome is preeminent because of its descent from St. Paul and St. Peter.
When we get on into the 2nd and early 3rd century, the references to the primacy of Peter are so numerous (Tertullian, St. Cyprian, St. Dionysius of Alexandria, etc.) that there's no point in contesting it.