That some sort of papal organizational, administrative, or legal bond among the churches could be derived, extrapolted from, or read into the First Epistle of Clement to the Corinthians is astounding to me. Unlike Stephen, Clement made no demand in his own name as the personal, exclusive successor of Peter based on his interpretation of Mathew 16:18,19 in the way that Stephen did. Stephen was the first to do so, not Clement.
When because of its prestige the church at Rome was asked for its view or advice in the solution of the Corinthian problem Clement replied not in his own name as a Pope of today would do, but in the name of his church. It begins, "The Church of God which is in Rome, to the Church of God which is in Corinth", and remains collective throughout. By my count, in the letter there are twenty nine citatations to the authority of Scripture and exactly zero to Clement's personal authority. What other ecclesiology can be derived from the letter expressly mitigates against a papal ecclesiology because in 44:3, the congregations had to give their approval in the appointment of prebyters:
1Clem 44:1 And our Apostles knew through our Lord Jesus Christ that there would be strife over the name of the bishop's office.
1Clem 44:2 For this cause therefore, having received complete foreknowledge, they appointed the aforesaid persons, and afterwards they provided a continuance, that if these should fall asleep, other approved men should succeed to their ministration. Those therefore who were appointed by them, or afterward by other men of repute with the consent of the whole Church, and have ministered unblamably to the flock of Christ in lowliness of mind, peacefully and with all modesty, and for long time have borne a good report with all these men we consider to be unjustly thrust out from their ministration.
1Clem 44:3 For it will be no light sin for us, if we thrust out those who have offered the gifts of the bishop's office unblamably and holily.
1Clem 44:4 Blessed are those presbyters who have gone before, seeing that their departure was fruitful and ripe: for they have no fear lest any one should remove them from their appointed place.
1Clem 44:5 For we see that ye have displaced certain persons, though they were living honorably, from the ministration which had been respected by them blamelessly.
1Clem 45:1 Be ye contentious, brethren, and jealous about the things that pertain unto salvation.
1Clem 45:2 Ye have searched the scriptures, which are true, which were given through the Holy Ghost;
1Clem 45:3 and ye know that nothing unrighteous or counterfeit is written in them. Ye will not find that righteous persons have been thrust out by holy men.
Firmilian, and other bishops may be disputing the institution of the papacy, but the very fact that they were doing it means that there was someone else who asserted it.
Without a doubt, but the breadth and severity of their outrage indicates the novelty of Stephen's interpretation of Matthew 16:18,19. It was an innovation to them, one not based on Apostolic tradition.
Stephen had condemned Cyprian as ‘false Christ, false apostle, and practicer of deceit,’ because he advocated re–baptism; and the Bishop of Carthage reciprocated in kind. Since the headship which Stephen claimed was unwarranted, by the example of St. Peter, he could not force his brethren to accept his views. Even worse, his judgment opposed the authentic tradition of the Church. The bishop of Rome, wrote Cyprian, had confounded human tradition and divine precepts; he insisted on a practice which was mere custom, and ‘custom without truth is the antiquity of error.’ Whence came the ‘tradition’ on which Stephen insisted? Cyprian answered that it came from human presumption. Subverting the Church from within, Stephen wished the Church to follow the practices of heretics by accepting their baptisms, and to hold that those who were not born in the Church could be sons of God. And finally, Cyprian urged that bishops (Stephen was meant) lay aside the love of presumption and obstinacy which had led them to prefer custom to tradition and, abandoning their evil and false arguments, return to the divine precepts, to evangelical and apostolic tradition, whence arose their order and their very origin.
In a letter to Cyprian, Firmilian endorsed everything the bishop of Carthage had said and added a few strokes of his own...Recalling the earlier dispute about the date of Easter, he upheld the practice of Asia Minor by commenting that, in the celebration of Easter and in many other matters, the Romans did not observe the practices established in the age of the Apostles, though they vainly claimed apostolic authority for their aberrant forms. The decree of Stephen was the most recent instance of such audacity, an instance so grave that Firmilian ranked Stephen among heretics and blasphemers and compared his doctrines and discipline with the perfidy of Judas. The Apostles did not command as Stephen commanded, Firmilian wrote, nor did Christ establish the primacy which he claimed...To the Roman custom, Firmilian, like Cyprian, opposed the custom of truth, ‘holding from the beginning that which was delivered by Christ and the Apostles.’ And, Firmilian argued, by his violence and obstinacy, Stephen had apostacized from the communion of ecclesiastical unity; far from cutting heretics off from his communion, he had cut himself off from the orthodox and made himself ‘a stranger in all respects from his brethren, rebelling against the sacrament and the faith with the madness of contumacious discord. With such a man can there be one Spirit and one Body, in whom perhaps there is not even one mind, slippery, shifting, and uncertain as it is?’
(Karl Morrison, Tradition and Authority in the Western Church (Princeton: Princeton University, 1969), pp. 31-32).
Cordially,
Clement I wrote his letter as a bishop of Rome over the head of the local bishop. Functionally, that is papacy. If he did not call his office that, and did not spend much time establishing his authority, then that only shows that his prerogatives were well understood. There is a veiled reference to papacy though in the recount of Aaron's primacy among the twelve tribes.
It is also incorrect to consider the authority of the pope "personal", -- to this day it is not, it is the authority of the office.
The "breadth and severity of [Firmilian's] outrage" reflects nothing but the disagreement over the rebaptism. Naturally, he throws everything and the kitchen sink at pope Stephen. Even so, the passages you cite point to the opinion that repabtism, -- not Stephen's authority -- is what Cyprian and Firmilian consider unapostolic.
Again, there are some today who disagree about Vatican II and they write, very convincingly, that we have no Pope, or that the Pope could not do what he in fact did, or that the Apostolic Church is not what we have. Agree or disagree with them on substance (the rebaptizers were, as a matter of fact, wrong), it has no bearing on the objective fact that the Catohlic Church is headed by the pope.