Yes, Matthew, this post is for you. 🙂
In The Fullness of Faith: On the Centrality of the Distinctively Catholic
Notwithstanding all the problems connected with the papacy throughout the history of the Church, two things speak in favor of its recognition within the Communio Sanctorum and its apostolicity.
In the first place (and we have already touched upon this) the Petrine element is taken for granted, so to speak, right at the beginning, in the Petrine texts of the New Testament. And of these the most impressive is not the passage in Matthew but rather the overpowering apotheosis of Peter at the end of John’s Gospel of love, which begins with the choosing of Peter in the first chapter and contains, at its center, the Apostle’s great confession of faith in the Lord.
The Office of Peter and the Structure of the Church
Just as Peter builds on John and John is within (and beside) Peter, the Petrine aspect perhaps appears nowhere more clearly than in Paul. Conversely, Pauline influence is unmistakable in Peter’s letters,  which are evidently intended to transmit wholly Petrine tradition. Again, we see two striking figures (who do not in the least blur each other and who have distinct theological and ecclesiological valences) in perichoresis, nor could it be otherwise among the members of the “living Body of Christ”.
Still, not every member communicates in the same way with the other. Within the manifest structure (which we stress is not definable in terms of tight distinctions) there are delicate lines of relation, most clearly drawn and represented by Luke and John. Luke portrays a family relationship between Mary and the Baptist, and, as Paul’s companion, he circumspectly builds a bridge between the latter and the Gospel tradition. Luke and John both bring to light deep, hidden mariological dimensions. In the episode at the foot of the Cross, told only by John, he who in the beginning of the Acts of the Apostles is always shown together with Peter becomes “son” and guardian of the Mother. Thus he is shifted into a discrete but totally indispensable central position (mediating between Peter and Mary, between the official, masculine Church and the feminine Church) that alone can give these two dimensions of the Church’s mysterium their place and proportion. Only where these concrete proportions are seen, understood and meditated upon in the light of faith, can one speak to advantage about the office of Peter in the Church. Moreover, this cannot be isolated from its most intimate connection with and within the collegium of the Twelve, each of whom was explicitly called by name.