One can grant your interpretation of the verse—and initial post too, of Peter being the leading disciple, without having to logically grant any sort of primacy to the Bishop of Rome.
Peter’s link to Rome is tenuous at best, as Linus was the first Bishop of Rome, NOT Peter, and Paul probably spent more time in Rome than Peter anyway—having first addressed his most ambitious epistle to Rome—and later being there for trial under Ceasar under house arrest.
The whole schema of trying to prove the permanent primacy of Rome via Peter came hundreds of years after Peter was martyred anyway. It’s clear it was a post-facto argument, by supporters of the early medieval world’s most powerful city’s Bishop.
Classic Roman Catholic “proofs” here, of out of context verses, and revisionist history. Oh well!
In essence, what I’m saying is that I RESPECT AND VENERATE Peter and his authority SO MUCH, that I reject the idea that that respect, veneration, and authority....are somehow passed on to Bishops of a particular ancient city.
NO Bishops of Rome have pretended to add to the Bible (though later, many claimed to have equal-authority to the Bible) and especially early on, in the Patristic period, the INCREDIBLE respect the Fathers had for the original Apostles, indicates they wouldn’t dream of thinking of the Bishop of Rome as as an equal to Peter.
NO ONE since the 1st Century has the authority of the Apostles—the New Testament being their full written testimony—a product of the Holy Spirit through the APOSTLES (NOT the later Church councils that recognized those writings). The Apostles themselves gave the qualification for the office of “Apostle” which was to be “a witness of His resurrection.” (Acts 1:22)
Arguments to the contrary—came MUCH later.