I disagree, as you well know. The RC tradition is self explaintory after you've been an RC and had it explained to you. It may be a bit of reflex to see all of that but scripture alone does not give you a pope at all. Just an ambiguous statement about a rock and another rock. That other rock, the profession not Peter, is the foundation. That interpretation fits with the rest of scripture. The pope explination does not.
Please. Where is the other rock? The episode starts with Peter's profession of faith, Christ in return calls him rock, and the foundation of the Church because of his faith, and give him the keys. Your interpretation is extra-scriptural.