Here's a quick Biblical reason why the Pope continues to exercise Peter’s authority:
In Isaiah 22, King Hezekiah has discovered that his household steward Shebna has been stealing money from the King. The obvious evidence is the pricey tomb Shebna has made for himself:
“What have you to do here and whom have you here, that you have hewn here a tomb for yourself, you who hew a tomb on the height, and carve a habitation for yourself in the rock?”
The King banishes his corrupt chief steward:
“Behold, the LORD will hurl you away violently, O you strong man. He will seize firm hold on you, and whirl you round and round, and throw you like a ball into a wide land; there you shall die, and there shall be your splendid chariots, you shame of your master's house.”
And makes a new one of Eliakim. He dresses him in the official clothes :
I will call my servant Eliakim the son of Hilkiah, and I will clothe him with your robe, and will bind your girdle on him, and will commit your authority to his hand; and he shall be a father to the inhabitants of Jerusalem and to the house of Judah.”
And gives him the key to the Kingdom, the House of David:
“And I will place on his shoulder the key of the house of David; he shall open, and none shall shut; and he shall shut, and none shall open…and he will become a throne of honor to his father's house.” Of course when Eliakim dies or falls out of royal favor, the King will get himself yet another prime minister- it’s not a one-time status unique to Eliakim.
Centuries later, Jesus borrows from this scene of a King authorizing his #1 official when he tells Peter:
“And I tell you, you are Peter, and on this rock I will build my church, and the powers of death shall not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven."
So Peter is entrusted with the keys not to an earthly kingdom, but the Kingdom of Heaven. Like Eliakim, he will be a father, a papa, to the people. And being a key-holder, when Peter dies or retires a new prime minister will take his place.