The reader below has pressed me further on my claim that the Bishop of Rome is the sole successor to the Petrine office. And rightfully so, for as I admitted below, I did not attempt to provide an explanation of why this is the case, except to show that everyone in the ancient Church seems to have assumed it was the case.
I would suspect that a proper and fuller answer to the question would involve the theology of sacred tradition, by which I mean not only the Church's oral, dogmatic tradition, but also the broader notion of tradition which incorporates all of salvation history, along the lines of the words of Pope John Paul II
quoted above, "the mystery of the divine plan which arranges the course of human events to serve the Church's beginnings and development." The Second Vatican Council
speaks of divine revelation occurring through not only the 'words' of Scripture and ecclesiastical teaching, but also the 'deeds' of salvation history:
"This plan of revelation is realized by deeds and words having in inner unity: the deeds wrought by God in the history of salvation manifest and confirm the teaching and realities signified by the words, while the words proclaim the deeds and clarify the mystery contained in them."
If we face the bare facts of the history of the early Church, there are a number of doctrines and ecclesial institutions that are of somewhat 'sketchy' origin. Would the doctrine of original sin have entered the Church's credo had Augustine, interpreting Romans 5:12, not been reading a faulty translation of the Latin New Testament? Would Eutychianism have triumphed in 450 if Emperor Theodosius II, intent on forcing it upon the universal Church, had not fallen off his horse and broken his neck? Would Pelagius have secured a formal commendation from Rome had his compatriots not burned down the monastery of the Pope's secretary days before the meeting?
Thus, in describing the origins of Petrine primacy in Rome, the Pope mentions
two central considerations:
"There is a negative one which, beginning with the need of a succession to Peter by virtue of Christ's very institution . . . confirms that there are no signs of such a succession in any other Church. Moreover, there is another consideration we could call positive: it consists in showing the convergence of signs that in every age point to the See of Rome as that of Peter's successor."
The 'convergence of signs' in 'every age' might include: (1) Christ's exclusive charge to Peter to serve as the foundation of His Church (Mt. 16:18) and to shepherd His sheep (Jn. 21:15-19), (2) Peter's journey to Rome and martyrdom there, along with his co-apostle Paul, (3) the political status of Rome as the sole capital of the Roman Empire, soon to dominate the known world, (4) the overwhelming devotion and respect which all Christians, over the next decades and centuries, eagerly offered to the See of Rome, and which led them to utilize the See as a final court of doctrinal appeal, and lastly (5) the clear realization among Christians that the unity of Christ's Church required a single See capable of preserving that unity, none being more capable or appropriate than that residing in the Eternal City.
The dogmas which the Church holds dear, by and large, were not delivered by an angelic messenger emblazoned on pristine parchment in glittering script. Certainly, all have some basis in Scripture, if only implicit and seminal. The concrete outworking of these beliefs requires a lot of nitty-gritty hard work, and this is why the Catholic Church has always put Sacred Tradition alongside Sacred Scripture. And, in my view, Tradition involves not only personally-conveyed oral truths, but the ambiguity and messiness of human history, with all of its unexpected twists and turns. It is always possible that a given event in history, which may seem arbitrary and even meaningless to an objective observer, may also carry the weight of divine providence, i.e. 'deeds wrought by divine providence in the history of salvation'. Of course, it is too often only in hindsight that these deeds can be recognized for what they are, if they are recognized at all.
In short, without having undertaken an exhuastive study of the origins of Petrine Primacy in Rome, I am quite open to the idea that much of it involved human industry, political maneuvering, and yes, historical accident. But my only point here is that it is precisely within these activities that the hand of God can be discerned. John Henry Newman, in his Essay on the Development of Christian Dogma, once referred to Christ's words to Peter in Matthew not as an indicative statement, but as a prophecy - a prophecy which would take decades, centuries to develop, in stops and starts, often in the most 'natural' of ways, seemingly imperceptible to those who lack the faith in this divine prophecy. But a prophecy which, due to the authority of He who gave it, could not fail to bear fruit in time, however inelegant the course of the tree's growth.