I think that both things are true. Certainly Peter’s confession of faith is a Rock. Jesus is also THE Rock, and at that moment Jesus grafted Peter into His own rockiness, changing his name to reflect this.
Are you ready to delve into the early fathers? This is where we see the Apostolic Succession really opened up. The NT Church seen in Acts still has the Apostles, and the Succession was not so critical until they began to depart to their heavenly reward.
This is a secular world view of leadership, and has no place in the Kingdom. Jesus made it clear that those in leadership would not take on the “supremacy” attidute prevalent in society. To interject the notion of it is contrary to Christ’s plan. On the contrary, the pope signs his letters “servant of the servants of God” fittingly. The primacy of the Christian Church in Rome is not based on it’s own claims, or any “supremacy” but in the adherance to right doctrine, upon which the unity of the whole Church is based:
“The blessed Apostles [Peter and Paul], having founded and built up the Church [of Rome], they handed over the office of the episcopate to Linus. Paul makes mention of this Linus in the Epistle to Timothy. To him succeeded Anencletus; and after him, in the third place from the Apostles, Clement was chosen from the episcopate. He had seen the blessed Apostles and was acquainted with them. It might be said that He still heard the echoes of the preaching of the Apostles, and had their traditions before his eyes. And not only he, for there were many still remaining who had been instructed by the Apostles. In the time of Clement, no small dissension having arisen among the brethren in Corinth, the Church in Rome sent a very strong letter to the Corinthians, exhorting them to peace and renewing their faith. To this Clement, Evaristus succeeded; and Alexander succeeded Evaristus. Then, sixth after the Apostles, Sixtus was appointed; after him, Telesphorus, who also was gloriously martyred. Then Hyginus; after him, Pius; and after him, Anicetus. Soter succeeded Anicetus, and now, in the twelfth place after the Apostles, the lot of the episcopate has fallen to Eleutherus. In this order, and by the teaching of the Apostles handed down in the Church, the preaching of the truth has come down to us.” (St. Irenaeus, Against Heresies, A.D. 180, [3,3,3])
Certainly not, but each one can do it in the light of the Holy Spirit. This is also true of infallibile teaching. No one can teach infallibly except by the HS. This is the model represented in scripture:
17 And Jesus answered him, "Blessed are you, Simon Bar-Jona! For flesh and blood has not revealed this to you, but my Father who is in heaven.
Teaching or confessing infallibly is not a work of flesh and blood, but of the Father, who reveals it by the HS. It is by grace, lest any man should boast.