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.
The Roman Catholic Church is headed by the pope, on that I'll agree.
Cordially,