The parallel of words gives the correlation to the key of the House of David, not some other keys. Jews of the day knew their Holy Scripture. The key of the House of David spoken of is the key given by the king to his prime minister to have authority when the king is gone. Someone had to handle the regular duties of the kingdom when the king was not present. The prime minister is not a king, not even royalty. But he is given a special job, in Isaiah it states to be a
father to the people. It was Peter alone, not any of the other Apostles, who was given the special authority:
Isaiah 22: 21-22 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. 22 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. RSVCE
Revelation 3:7 “And to the angel of the church in Philadelphia write: ‘The words of the holy one, the true one, who has the key of David, who opens and no one shall shut, who shuts and no one opens RSVCE
Never depicted being the head of the earthly Church? Jesus changed his name from Simon to Rock, and Rock was often used to refer to God. In John 20 we see John deferring to Peter to enter the tomb. There are a number of times where Peter speaks for the Apostles. Or where the Bible refers to Peter and the rest of the Apostles. And there is more:
Jesus and Peter
15 When they had finished breakfast, Jesus said to Simon Peter, “Simon, son of John, do you love me more than these?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Feed my lambs.” 16 A second time he said to him, “Simon, son of John, do you love me?” He said to him, “Yes, Lord; you know that I love you.” He said to him, “Tend my sheep.” 17 He said to him the third time, “Simon, son of John, do you love me?” Peter was grieved because he said to him the third time, “Do you love me?” And he said to him, “Lord, you know everything; you know that I love you.” Jesus said to him, “Feed my sheep.[
a] RSVCE