…] Just that there is one in particular who rules over all
You are, of course, entitled to this opinion; is this how the Fathers have understood this event, however? It doesn’t appear to be so. From
St. Cyril’s commentary on Luke, we learn the following:
*Now it was from an unprofitable love of glory, the root of which is pride, that this vain and senseless ambition had, so to speak, shot up. For the very fact of wishing at all to be sot over others, and to strive for this end, renders a man liable to be justly blamed: though, on the other hand, it is not absolutely destitute of that which may fitly be praised. For to be exalted in virtue is worthy of all estimation: but those who would attain to it must be of modest mind, and possess such humbleness of feeling as to abandon out of love to the brethren all idea of preeminence. And such the blessed Paul would also have us be, thus writing, “Consider as regards your companions, that in honour they are better than you.” For so to feel is highly worthy of the saints, and renders them glorious, and makes our piety to God more worthy of honour: it tears the net of the devil’s malice, and breaks his manifold snares, and rescues us from the pitfalls of depravity: and finally, it perfects us in the likeness of Christ the Saviour of us all. For listen how He sets Himself before us as the pattern of a humble mind, and of a will not set on vainglory: for "Learn, He says, of Me, Who am meek and lowly in heart. …] The Saviour then forewarned him [Peter] what would have been the result had he been yielded up to Satan’s temptation: but at the same time He offers him the word of consolation, and says, “And do you also hereafter, when converted, strengthen your brethren:” that is, be the support, and instructor and teacher of those who draw near to Me by faith. And moreover, admire the beautiful skill of the passage, and the surpassing greatness of the divine gentleness! For, lest his impending fall should lead the disciple to desperation, as though he would be expelled from the glories of the apostleship, and |677 his former following (of Christ) lose its reward, because of his proving unable to bear the fear of death, and denying Him, at once Christ fills him with good hope, and grants him the confident assurance that he shall be counted worthy of the promised blessings, and gather the fruits of steadfastness. For He says, “And do you also, when converted, strengthen your brethren.” O what great and incomparable kindness! The disciple had not yet sickened with the malady of faithlessness, and already he has received the medicine of forgiveness: not yet had the sin been committed, and he receives pardon: not yet had he fallen, and the saving hand is held out: not yet had he faltered, and he is confirmed: for “do you, He says, when converted, strengthen your brethren.” So to speak belongs to One Who pardons, and restores him again to apostolic powers.
But Peter, in the ardour of his zeal, made profession of steadfastness and endurance to the last extremity, saying that he would manfully resist the terrors of death, and count nothing of bonds; but in so doing he erred from what was right. For he ought not, when the Saviour told him that he would prove weak to have contradicted Him, loudly protesting the contrary; for the Truth could not lie: but rather he ought to have asked strength of Him, that either he might not suffer this, or be rescued immediately from harm. But, as I have already said, being fervent in spirit, and warm in his love towards Christ, and of unrestrainable zeal in rightly performing those duties which become a disciple in his attendance upon his Master, he declares that he will endure to the last extremity: but he was rebuked for foolishly speaking against what was foreknown, and for his unreasonable haste in contradicting the Saviour’s words. For this reason He says, “Verily I tell you, that the cock shall not crow to-night, until you have thrice denied Me.” And this proved true. Let us not therefore think highly of ourselves, even if we see ourselves greatly distinguished for our virtues: rather let us offer up the praises of our thanksgivings to Christ Who redeems us, and Who also it is that grants us even the desire to be able to act rightly: by Whom and with Whom to God the Father be praise and dominion, with the Holy Spirit, for over and ever, Amen.*
So it seems that the Fathers understood the “singling out” of St. Peter to be a means by which he was restored to apostleship in light of his great error in denying Christ. St. Cyril does call him “the foremost”, but in context of the sermon it would be incredibly ridiculous to assert that this shows that St. Peter’s successors in Rome have temporal power or final say over any others, when St. Cyril repeats over and over that such an ambitious idea is unprofitable and not fitting the apostles’ mission (and if St. Cyril says this of the apostles, I very much doubt he would change his tune for any modern patriarch, whether in Rome or anywhere else).