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.
I’m wasn’t speaking of apostolic succession, but papal supremacy, but I see how you can apply that verse to either concept. I just don’t think that is correct. The Bible says that all who believe in Christ are part of the body of Christ, so they are all part of this church. This must include people who died before the Catholic church existed, who could never have darkened the doors of one of its franchises. Therefore, the church spoken of as the Body of Christ can’t be limited to one subset of it, even if that subset has an impressive vintage.