Christ did give the mantle of leadership to Peter. That much is not in question. Peter took over as head of the Church. That’s the way Jesus set it up.
However, Peter is not the rock Christ is referring to. Jesus is the firm, never moving rock. Not some fallible man.
The real problem with Catholicism isn’t Peter, it’s Linus I.
After Peter died Linus claimed to be the head of the Church. Only problem is, there was still a living Apostle on the earth during this time; John. John was next in line and was the true leader of the Church not Linus.
Linus didn’t continue Christ’s Church, he created his own in the eyes of God.
Even while Linus was claiming to lead the church, Jesus appeared to John and spoke to him face to face. Jesus never appeared to Linus ever. And that’s a fact.
Jesus’ appearance to John proves John’s authority in the eyes of God.
After Peter died Linus claimed to be the head of the Church
Actually, there is no record of Linus claiming to be anything. What is recorded is that in 180, Irenaeus wrote that the apostles committed to Linus the office of the episcopate.
Well, Peter was fallible, And the point that Irenaeus makes is The fidelity of the Roman Church to the apostolic tradition, and of its bishops as keepers of the keys. That admits the confusion of competing view in the 2nd Century. From the first, the Church haas been subject to “heresies,” of individual opinions that divide Christians. The search is for that doctrine that comes from the apostles and is the same everywhere. Rome acquired great prestige from its association with Peter and Paul, but it is the authority of Peter that matters more. But it is unnecessary to think of Peter as a kind of prelate, His charism is what is seen in the first half of Acts.