I am a former-Catholic who happens to believe that the papacy is a historical fiction that wasn't present in the earliest church. Even with that bias, I can plainly see that this is just downright silly.
If I understand correctly, you are technically correct. If Peter was really the Apostle-in-Chief of Rome, as I believe he was, be was the bishop of an underground church. Constantines declaration of the church as the state religion of Roma made it, for better or worse, the truly Catholic religion.