Actually, the Catechism says that Christ would build His church on Peter's confession. Not Peter.
From the Catechism of the Catholic Church
http://www.vatican.va/archive/ENG0015/_INDEX.HTM
424 Moved by the grace of the Holy Spirit and drawn by the Father, we believe in Jesus and confess: “You are the Christ, the Son of the living God.”8 On the rock of this faith confessed by St. Peter, Christ built his Church.9 (683, 552)
It is very clear that Christ appointed Peter as the first Pope and the Apostles as the first bishops. But the main point of my comment is the Jesus has told us not to fear the threats, but rather to persevere because in the end His church will triumph.
552 Simon Peter holds the first place in the college of the Twelve;283 Jesus entrusted a unique mission to him. Through a revelation from the Father, Peter had confessed: “You are the Christ, the Son of the living God.” Our Lord then declared to him: “You are Peter, and on this rock I will build my Church, and the gates of Hades will not prevail against it.”284 Christ, the “living stone,”285 thus assures his Church, built on Peter, of victory over the powers of death. Because of the faith he confessed Peter will remain the unshakeable rock of the Church. His mission will be to keep this faith from every lapse and to strengthen his brothers in it.286 (880, 153, 442, 424)
553 Jesus entrusted a specific authority to Peter: “I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven.”287 The “power of the keys” designates authority to govern the house of God, which is the Church. Jesus, the Good Shepherd, confirmed this mandate after his Resurrection: “Feed my sheep.”288 The power to “bind and loose” connotes the authority to absolve sins, to pronounce doctrinal judgments, and to make disciplinary decisions in the Church. Jesus entrusted this authority to the Church through the ministry of the apostles289 and in particular through the ministry of Peter, the only one to whom he specifically entrusted the keys of the kingdom. (881, 1445, 641, 881)
And as you know, the context of Matthew 16:18 is that the Rock not Peter, but Christ.
Just five verses after Matthew 16:18, Christ rebukes Peter: "But he turned, and said unto Peter, Get thee behind me, Satan: thou art an offence unto me: for thou savourest not the things that be of God, but those that be of men." (Matthew 16:23). Hard to imagine Christ's church being built on a human foundation in that light.
And of course, Peter himself confesses that Christ is the "chief cornerstone" (1 Peter 2:6).