I know you said you're done here more than once but please take just one more moment to consider something.
From early in the OT (in fact, first promised to Abraham), God promised an everlasting kingship out of Israel over the whole world. Later, that kingdom is defined as on the order of King David.
When Solomon assumed the throne after David, the first things he did were to enthrone his own mother as the queen mother to hear the supplications of his people and established his ministers. If I could point you to chapter and verse, I would but I don't know them off the top of my head. What you will find, if you take the time, in the institution of the Prime Minister is nearly the exact wording Jesus said to Peter of binding and loosing and the symbol of his authority were the keys to the kingdom.
The Catholic Church is very consistent in its interpretation of Peter's office and responsibility. If Jesus is a king of the Davidic line it isn't unreasonable to see His institutional words as they were understood through Solomon's wisdom.