We aren't talking about history, we're talking about Christianity. The office of "Pope" or "Vicar of Christ" is a late invention. It didn't exist at the time of Christ. It was not ordained by Christ. It had nothing whatever to do with the ministry of the Apostles. And it is nowhere in the Christian Covenant.
Covenant. Know what the word means? It's a legal term. God wrote the Covenant and sealed it in Christ's blood, handing it to us on a 'take it or leave it' basis. There is no editorial authority given to men. None.
Now, you can tell us about history; but, all that tells us is that a bunch of men usurped athority they do not and did not possess and have no right to wield under the covenant.
They did the same thing with the Canon of Scripture - usurping the authority of the Jews to Whome Paul notes plainly were given authority over the Oracles of God - the Old covenant in other words. So it would seem we have to define our terms, huh. Christian means one thing when reading scripture and is twisted into quite another by "history". So for this conversation, I think we need to look at the authoritative speaker with regard to the issue - that would be scripture. And scripture makes no mention of any office of "Pope" aka "vicar of Christ". Nowhere. Doesn't exist and is a Catholic Usurpation.
Anything else?
Also the New Testament doesnt mention the word Trinity either buts the reality of the Trinity is in the scriptures. I mean we can have this argument all night. Catholics have biblical proofs for the Papacy as well as proofs from Holy Tradition.
The Catholic Church gets its legitimacy from Apostolic succession. Something a Church of Christ member (which you obviously are) couldn't possibly understand...