I agree. But that is true because although he tempted Christ, Peter didn't profess a different faith.
Our Lord said "Thou art Peter and upon this rock I will build my Church ....", He did not say, "....on this rock I will build my Church, if you ever lose the faith, or, ...if you ever preach heresy, or ....if you ever start another church, you lose your office ipso facto". He never said that and it is not implied anywhere.
Applying basic Catholic principles, what Our Lord told St. Peter when He made St. Peter the first pope is, basically: "Peter, you're it, I am making you the supreme authority on earth, answerable only to Me - you and your successors knowing that salvation is only possible in my Church, you and your successors had better do your jobs and feed my sheep, or you've had it come your judgement day".