Is Peter the "Petra" of Matthew 16?

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As the Pope makes his visit to America, I find it important to bring some clarity to the subject.  One thing that is sadly missing is any theological significant discussions of the doctrine of the papacy itself.*  Significant in the Roman Catholic system is the understanding that the Pope is the Vicar of Christ, the head of the true Church, who infallibly can define doctrine.  They believe that the Pope has taken the “seat of Peter”, who they believe to be the Prince and Chief of the apostles, and that those who follow after him as the Bishop of Rome.  This doctrine was first clearly articulated in the Vatican 1 Council.  In this council the church infallibly interpreted Matthew 16:13-20 as well as John 21 to refer to Peter’s primacy.  They also made the claim that this has been the teaching of the church since the beginning.  Vatican 1 reads:

We teach and declare that, according to the gospel evidence, a primacy of jurisdiction over the whole Church of God was immediately and directly promised to the blessed apostle Peter and conferred on him by Christ the lord.

It was to Simon alone, to whom he had already said You shall be called Cephas , that the Lord, after his confession, You are the Christ, the son of the living God, spoke these words:

Blessed are you, Simon Bar-Jona. For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are Peter, and on this rock I will build my Church, and the gates of the underworld shall not prevail against it. 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.

And it was to Peter alone that Jesus, after his resurrection, confided the jurisdiction of Supreme Pastor and ruler of his whole fold, saying: Feed my lambs, feed my sheep.

To this absolutely manifest teaching of the Sacred Scriptures, as it has always been understood by the Catholic Church, are clearly opposed the distorted opinions of those who misrepresent the form of government which Christ the lord established in his Church and deny that Peter, in preference to the rest of the apostles, taken singly or collectively, was endowed by Christ with a true and proper primacy of jurisdiction…

And so, supported by the clear witness of Holy Scripture, and adhering to the manifest and explicit decrees both of our predecessors the Roman Pontiffs and of general councils, we promulgate anew the definition of the ecumenical Council of Florence, which must be believed by all faithful Christians, namely that the Apostolic See and the Roman Pontiff hold a world-wide primacy, and that the Roman Pontiff is the successor of blessed Peter, the prince of the apostles, true vicar of Christ, head of the whole Church and father and teacher of all Christian people.

 That apostolic primacy which the Roman Pontiff possesses as successor of Peter, the prince of the apostles, includes also the supreme power of teaching. This Holy See has always maintained this, the constant custom of the Church demonstrates it, and the ecumenical councils, particularly those in which East and West met in the union of faith and charity, have declared it. -Vatican 1, Session 4, 1:1-4, 3:1 4:1 (https://www.ewtn.com/library/COUNCILS/V1.HTM)

The Roman doctrine of the papacy stands or falls on Matthew 16:13-20.  We will examine this Roman Catholic dogma in three phases: first, looking at Matthew 16:13-20, second, looking at relevant New Testament passage, and third, looking at history before giving concluding thoughts.