You left out the part where Peter denied his Lord three times. If that was the rock to be the foundation of a church, they were in trouble. Fortunately we know Christ is the Foundation. He is the cornerstone not Peter. These claims of apostolic succession carry little weight for me because I find no scripture making claims this was the way it was supposed to be. In fact I can’t find the word Pope in the Bible.. why is that?
Only if we, as some seem to believe, thought that it was anything other than the grace of God that made Peter fit for the job. Despite what people keep on saying about us, we think it's the work of God that keeps Popes from messing up any worse than they do, just as we believe it is the Spirit of God which makes the Church Christ's body. It's certainly not any merit or particularly excellent character intrinsic to the members of the Church.
The Catholic church has regularly tried to justify the existence of their doctrinal stances from biblical verses that in no way apply to their beliefs. But that’s where they rely more on church teachings than they do the holy scriptures.