But the true Church will still have Peter
Peter is the key. Jesus himself promised to give him the keys of the kingdom in Matthew 16:19. The keys of the kingdom is a symbol of dynastic authority in Davidic kingdoms. We see this in Isaiah 22:15-25 which uses remarkably identical language to Matthew 16:18-19
Thus says the Lord GOD of hosts, “Come, go to this steward, to Shebna, who is
over the household, and say to him: 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? 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.I will thrust you from your office, and you will be cast down from your station. In that day I will call my servant Eli’akim the son of Hilki’ah, 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 (pope = papa = father!

). 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 I will fasten him like a peg in a sure place, and he will become a throne of honor to his father’s house.
And they will hang on him the whole weight of his father’s house, the offspring and issue, every small vessel, from the cups to all the flagons. In that day, says the LORD of hosts, the peg that was fastened in a sure place will give way; and it will be cut down and fall, and the burden that was upon it will be cut off, for the LORD has spoken.”
“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**.”
When Satan wanted to sift all of the apostles, Jesus prayed for Peter alone.
“Simon, Simon, behold Satan has demanded to sift
all of you like wheat, but I have prayed that
your own faith may not fail; and once you have turned back, you must
strengthen your brothers.” (Luke 22:31-32)
In fact, the Pope recognizes that his ministry as the successor of Peter is to strengthen his brethren.
“The approval and publication of the Catechism of the Catholic Church represents a service which the
Successor of Peter wishes to offer to the Holy Catholic Church, and to all the particular Churches in peace and communion with the Apostolic See: the service, that is, of
supporting and confirming the faith of all the Lord Jesus’ disciples (cf. Lk 22:32), as well as of strengthening the bonds of unity in the same apostolic faith.”