Since you mention Clement, here he speaks with authority....
Oh, I have no doubt he speaks with authority. And he may have well been a bishop. But he never identifies himself as an apostle. His statement to the effect of things "which have been said by [God] through us" may well have been a reference to things previously said or things said to the church had already established. He never, to my knowledge, ever claimed that God's will came through him as a senior apostle.
After Peter, Linus held the bishopric from 67-79 A.D. Then Cletus, who served from 79-91. He was succeeded by Clement I, who presided from the 91-100. These men were most likely were legitimate bishops of Rome, but nowhere do we read of any revelations they received, nor that they were apostles. John, on Patmos, had previously been promised by Christ that he would live to see the Second Coming of Christ (John 21:20-23), so he would have been the senior apostle until he was removed from his terrestrial duties. According to "guesstimates," he wrote the Apocalypse around 90 A.D., which means he was very old, and would easily have been the senior apostle by that time. (Note: John was not told he would not die, but that he would "live" or "tarry til I come." All men die, but Elijah and John were given extended lives.)
From the Shepherd of Hermas: "Therefore shall you [Hermas] write two little books and send one to Clement [Bishop of Rome] and one to Grapte. Clement shall then send it to the cities abroad, because that is his duty" (The Shepherd 2:4:3 [A.D. 80]).
There is much of value in the Shepherd of Hermas, but little is known of its authorship and it was never canonical. We also don't know to the extent it was tampered with. It is said to have been written by Hermas, the brother of Pius, who served as bishop beginning in 142 A.D. A.D. Howell-Smith wrote of it: "The theology of the Church must have been very elastic at a time when such a book could enjoy popularity and implicit, if not explicit, ecclesiastical sanction, for its Christology does not seem to square with any of the Christologies of the New Testament, or with those of contemporary theologians whose occasional documents have reached us." (Jesus Is Not a Myth, ppgs. 120-121) Pius is considered by many to be the first "pope," and The Shepherd is an allegory, not a history.
You also quote Tertullian's view of the keys of the kingdom: "For though you think that heaven is still shut up, remember that the Lord left the keys of it to Peter here, and through him to the Church, which keys everyone will carry with him if he has been questioned and made a confession [of faith]" (Antidote Against the Scorpion 10 [A.D. 211]).
But what does he mean "though you think that heaven is still shut up"? To me, it proves what I've been saying. The gifts of the Spirit get lost as the church goes deeper into apostasy. The opinion he's addressing is those who have realized that revelation has ceased, and he's saying, "Hey, no, we've still got the keys of the Kingdom if nothing else!" But by that time the apostasy, in my view, had set in. The body was still warm, but people had begun noticeing that something wasn't right with the church. There were no more visions. God had apparantly ceased speaking to it. And as Origen said, "Where there is no revelation, there is no certitude." As Dr. Hugh Nibley, admittedly a proponent of the apostasy, said: "The formal ecstasies and intellectual insights of the schoolmen are not real revelation, and Augustine knew it. In all fairness to him we must report that he would infinitely have preferred revelation to philosophy. Not only did he feel guilty about what he was doing, but it was only after long years of agonizing struggle and indecision that he at last, painfully and with heartbreaking reluctance, closed the book on revelation or recognized that he could not open it. What a difference there is, he cries in the City of God, between the ambiguities of the academicians and the certainty of the Christian faith!" (World and the Prophets, 3rd Ed., ppgs 91-92, citing Augustine, The City of God XIX, 18, in PL 41:646).
He goes on to say, "In the 270 letters of Augustine that have survived, we see the man at work trying to answer the great questions of doctrine and administration that should have been answered by the head of the church. Letters pour in to him from all over the Christian world, and he answers them as best he can: He never refers the questioners to any higher authority, even though the cases are sometimes very serious and have nothing at all to do with his diocese; nor does he personally ever appeal to any higher authority, either in administrational or in doctrinal matters, however important they may be. This is not surprising if one knows the situation. 'If there had been, in the Church of the 4th century, a central authority recognized and active, it would have offered a means of solution. But it was not so.' Thus wrote Monsignor Duchesne, speaking of the administrative solution." (Citing The Early History of the Church (London: J. Murray, 1931), 2:521)
Oh, and by the way, Roman Catholics aren't the only church with Apostolic lineage.
Yes, I've taken the Orthodox orientation and to be honest, the Father teaching the course was even more critical than I am. Unfortunately, where there is no vision, the people perish. (Proverbs 29:18)