That "empirical fact" is not a fact at all, but is open to dispute. Clement's letter to the Corinthians commands their obedience, to him. And the parties in the Quartodeciman controversy turned to Rome to settle it. By the time you get to Chalcedon, the Papacy clearly has some sort of unique authority. The main question is whether that authority is ordinary, or only exercised in extraordinary circumstances.
As for his other "empirical facts," they all sidestep the only question that ought to matter: was the office of the Roman Pontiff instituted and intended by Christ? If it was, what difference does it make that it was occupied by corrupt men at some point in history?
The office of the High Priest was unquestionably instituted by God through Moses. Were all the Jewish High Priests good and righteous men? Hardly, but that didn't invalidate the office at all.
But as Lurking notes, we clearly live rent-free in your head. I'll keep praying for you.
That's not right. You should read the letter yourself. My favorite parts include the phoenix being real and salvation through faith.
The letter offered unsolicited advice. Since no one said he shouldn't have sent it, it is therefore assumed that he had the right to send it. Hence, he was over them and therefore supreme.