I agree with Kabane, I would only like to precise something.
St Leo, in his Tome, refers to Peter:
>Wherefore we all, in the very Creed, confess that “the only-begotten Son of God was crucified and buried,” according to that saying of the Apostle, “for if they had known it, they would not have crucified the Lord of Majesty.” But when our Lord and Saviour himself was by his questions instructing the faith of the disciples, he said, “Whom do men say that I the Son of Man am?” And when they had mentioned various opinions held by others, he said, “But whom say ye that I am?” that is, “I who am Son of Man, and whom you see in the form of a servant, and in reality of flesh, whom say ye that I am?” Whereupon the blessed Peter, as inspired by God, and about to benefit all nations by his confession, said, “Thou art the Christ, the Son of the living God.” Not undeservedly, therefore, was he pronounced blessed by the Lord, and derived from the original Rock that solidity which belonged both to his virtue and to his name, who through revelation from the Father confessed the selfsame to be both the Son of God and the Christ; because one of these truths, accepted without the other, would not profit unto salvation, and it was equally dangerous to believe the Lord Jesus Christ to be merely God and not man, or merely man and not God.
Peter's confession of faith is that Jesus is truly God and truly man. -As a consequence- Peter is the rock upon which the Church is built.
And because the Pope follows Peter's confession of faith, the attributes of Peter are his - he really follows the orthodox faith, he is really the successor and even the living presence of Peter:
>This is the faith of the fathers, this is the faith of the Apostles. So we all believe, thus the orthodox believe. Anathema to him who does not thus believe. Peter has spoken thus through Leo. So taught the Apostles. Piously and truly did Leo teach, so taught Cyril. Everlasting be the memory of Cyril. Leo and Cyril taught the same thing, anathema to him who does not so believe. This is the true faith. Those of us who are orthodox thus believe. This is the faith of the fathers. Why were not these things read at Ephesus [i.e. at the heretical synod held there]? These are the things Dioscorus hid away.
And this faith of Peter not only determines that the Pope is orthodox, but also that the council is ecumenical. Indeed, by this proclamation after the reading of the Tome, the council proclaims that its intention is to teach correctly as to what pertains to the humanity and the divinity of Jesus, and we find that this is the concern of the first 7 ecumenical councils: the Trinity (for the first 2) and the Incarnation (for the latter 5). Therefore Peter is relevant not simply for the Pope, not simply for the ecumenical council, but for both.
But, later Catholic ecumenical councils tend to either be about pastoral rather than dogmatic issues, or be about dogmatic issues that do not concern the nature of who Jesus is. There is a noticeable discontinuity, in my opinion, and the proud proclamation of faith that Jesus is truly God and truly man ceases to be the main doctrinal focus after the 7th ecumenical council.
Another thing I want to precise: a Catholic objection could be "well, either way, the Fathers agree that the Pope is necessay, and you don't have him anymore, so either way you cannot claim legitimacy". But is that really so? First, what decides our salvation is not a checklist of statements to agree with, but the sacraments, most principally the Eucharist. If being in communion with the Pope is necessary for salvation in a way that is distinct from being in communion with your own bishop, that would imply the Pope is not ordained like other bishops, or that there is a 8th sacrament of the "papal ordination", but that is not the case. One could reply then that because the Pope has immediate universal jurisdiction, he is "your" bishop just as much as your local bishop is "your" bishop, but as Kabane says, we would disagree with this claim. In what sense is the Pope necessary for the Church then? In the Fathers such as St Cyprian and St Maximus, we find that the Apostolic See is necessary because 1) it is a proof of unity among the Churches, so that communion with the Pope is proof of belonging to the Catholic communion of faith, and 2) it is a proof of continuity with the apostolic church: the Pope holds the position of Peter and the college of bishops holds the position of the other apostles. What happens if there is no Orthodox Pope then? The Church is gravely wounded, it is not functioning as it should, this is a grave non-canonical situation, but it does not prevent the salvation of its flock or endanger the sacraments.