I am not equating binding and loosing with opening and shutting. I am comparing the language. My point is that when Jesus said the words of Matthew 16:18-20, He said it in such a way that everyone present would think of Isaiah 22:15-23. Isaiah 22:15-23 is all about a vicar, a vice-president, a second in command, a royal steward - a specific office in the Kingdom of Israel.
My question to you - Why would Jesus do this? Why would Jesus make people think of Isaiah 22:15-23? You answer those questions and you will understand the Papacy.
The concept is the same. The keys to David’s house gave authority and access to the house of David. The “keys” gave access to the saving knowledge of Jesus (of David’s house) which is the gospel for salvation. The “keys” were the knowledge and access to Jesus who is the salvation of those who believed. And no it has nothing to do with the papacy. The Apostles gave everything we need with what they wrote about the saving grace of Jesus and the sacrifice He offered “once for all”. If anything all pastors, teachers, and leaders who bring the gospel message have the keys to heaven. It’s the “good news” of the gospel of Jesus Christ.