“A false dichotomy. Christ said to Peter, “And on this rock I will build MY Church.” Christ’s church IS Peter’s church. That’s what Christ said.”
Yes, yes, we’ve all heard the Catholic interpretation of that verse a thousand times already. If that was sufficient to convince Christians of the authority of the Pope to stand in the place of Christ, then it would have happened already. The fact is, not even all of the original churches planted by the Apostles hold to that interpretation, so why should I believe the private Roman interpretation?
The exact nature of a Petrine ministry -- a ministry given to Peter by Christ after the Resurrection, in the service of unity, of "confirming the brethren" -- has been argued about, and is argued still. But even at that, the Petrine Ministry question did not divide the church in schism for the first thousand years. The schism that then lamentably ensued did not disrupt the Petrine Succession; and both of the ecclesial communities (Catholic and Orthodox) divided by this schism, still carry on Apostolic Succession --- mutually recognized between them --- to this day.
That's no priate interpretation. It's worldwide belief and practice for 2,000 years, which we received from the APostles, whose whole authority is from Christ.
And if it's not public, then the word "public" has no meaning. It's the most public, manifest, and continuous Christian teaching, praying, governing body in the history of the world.