The argument for Papal authority over other Sees is largely one of necessity:
someone has to be the final authority in the event of dissent and disagreement on dogma, so let it be the successor of Peter, who held the primacy position among the Twelve. We see this in Irenaeus'
Adversus Haereses Book III, 3.2 ("For it is a matter of necessity that every Church should agree with this Church, on account of its pre-eminent authority
"). We see it in Pope Leo XIII’s Encyclical
Satis Cognitum (1896), who acknowledged apostolic authority separate from Rome, but denounced that authority as invalid if it is in conflict with Rome:
“For He who made Peter the foundation of the Church also ‘chose, twelve, whom He called apostles’ (
Luke vi., 13); and just as it is necessary that the authority of Peter should be perpetuated in the Roman Pontiff, so, by the fact that the bishops succeed the Apostles, they inherit their ordinary power, and thus the episcopal order necessarily belongs to the essential constitution of the Church. Although they do not receive plenary, or universal, or supreme authority, they are not to be looked as vicars of the Roman Pontiffs; because they exercise a power really their own, and are most truly called the
ordinary pastors of the peoples over whom they rule.
“But since the successor of Peter is one, and those of the Apostles are many, it is necessary to examine into the relations which exist between him and them according to the divine constitution of the Church. Above all things the need of union between the bishops and the successors of Peter is clear and undeniable. This bond once broken, Christians would be separated and scattered, and would in no wise form one body and one flock. ‘The safety of the Church depends on the dignity of the chief priest, to whom if an extraordinary and supreme power is not given, there are as many schisms to be expected in the Church as there are priests’ (S. Hieronymus,
Dialog, contra Luciferianos, n. 9). It is necessary, therefore, to bear this in mind, viz., that nothing was conferred on the apostles apart from Peter, but that several things were conferred upon Peter apart from the Apostles. St. John Chrysostom in explaining the words of Christ asks: ‘Why, passing over the others, does He speak to Peter about these things?’ And he replies unhesitatingly and at once, ‘Because he was pre-eminent among the Apostles, the mouthpiece of the Disciples, and the head of the college’ (Hom. lxxxviii.
in Joan., n. I). He alone was designated as the foundation of the Church. To him He gave the power of binding and
loosing; to him alone was given the power of
feeding. On the other hand, whatever authority and office the Apostles received, they received in conjunction with Peter.”
With all due respect to Leo, (1)
all of the apostles were given the power of binding and loosing (Matt. 18:18); (2) feeding lambs, if Peter alone was told to do it, has nothing to do with resolving disputes among the episcopacy.
You are right about ante-Nicene history. Most schisms and disputes of that era arose in the East, and they were resolved by bishops in the East. Rome played little if any role. If the early Popes had the "final say," they generally kept their mouths shut (assuming they were even consulted) about most heresies and disagreements. And sometimes when they did weigh in, they were told to go pound sand. In the middle of the third century, Pope Stephen’s view regarding the efficacy of baptism by heretics was rejected by 87 bishops at a Council of Carthage, at which Cyprian stated: “For neither does any of us set himself up as a bishop of bishops, nor by tyrannical terror does any compel his colleague to the necessity of obedience; since every bishop, according to the allowance of his liberty and power, has his own proper right of judgment, and can no more be judged by another than he himself can judge another.”
CHURCH FATHERS: On the Baptism of Heretics (Council of Carthage)
