I have seen on websites this and others trading quotes on church fathers favoring both arguments. What you can’t show me is any canon of an ecumenical council supporting your argument. The Ecumenical pre schism councils are infallible, Fathers not necessarily. Also again I believe Ignatius followed Peter in his first Bishopric at Antioch,therefore the current patriarch is also a successor of Peter. Scripture doesn’t mention Rome or state that only one bishop is Peter’s successor. the Bishop of Rome was first in Honor but did not exert authority over the whole chuch. It doesn’t matter what we want it matters what the church did historically. Again none of the early councils were held in Rome. And the bishop of Rome as well as all the other orthodox (small o) had to approve actions of the councils for them to be considered ecumenical.
The Synod of Alexandria, at which Athanasius was present, in its letter to Felix II., uses remarkable words on this point, and amongst other things, relates that, in the Council of Nice it had been determined that it was not lawful to celebrate any Council without the consent of the Holy See of Rome, but that the 'canons which had been made to that effect had been burnt by the Arian heretics. And in fact, Julius I., in the Rescript against the Orientals in Favour of Athanasius (CC. 2, 3), cites two canons of the Council of Nice which relate to this matter, which work of Julius I. has been cited by Gratian, four hundred years ago, and by Isidore nine hundred: and the great Father, Vincent of Lerius, makes mention of it a thousand years back. I say this because all the canons of Nice are not in existence, only twenty remaining: but so many grave authors cite others beyond the twenty, that we are obliged to believe what is said by those good Fathers of Alexandria above-named, that the Arians have got the greater part destroyed. For God’s sake let us cast our eyes on ancient and pure Church of the first six centuries, and regard it from all sides. And if we find it firmly believes that the Pope was successor of S. Peter, what rashness will it be to deny it?
This, methinks, is a reason which asks no credit, but pays in good coin. S. Peter has had successors in his vicarship: and who has ever in the ancient Church had the reputation of being successor of S. Peter, and head of the Church, except the Bishop of Rome? In truth all ancient authors, whosoever they be, all give this title to the Pope, and never to others.
And how then shall we say it does not belong to him ? Truly it were to deny the known truth. Or let them tell us what other bishop is the head of the Church, and successor of S. Peter. At the Council of Nice, at those of Constantinople and Chalcedon, it is not seen that any bishops usurp the primacy for himself: it is attributed, according to ancient custom, to the Pope; no other is named in equal degree. In short, never was it said, either certainly or doubtfully, of any bishop in the first five hundred years that he was head or superior over the rest, except of the Bishop of Rome; about him indeed it was never doubted, but was held as settled that he was such. On what ground, then, after fifteen hundred years passed, would one cast doubt on this ancient tradition? I should never end were I to try to catalogue all the assurances and repetitions of this truth which we have in the Ancients’ writings: but this will suffice just now to prove that the Bishop of Rome is the successor of S. Peter, and that S. Peter was and died Bishop at Rome.
(The Authority of the Pope, chapter X, Saint Francis de Sales (1567-1622)
catholicapologetics.info/apologetics/general/authorope.htm#CHAPTER_X