Again, I will ask you to please cite scriptural passages to support your statement.
In Matt. 16:18 - Jesus said in Aramaic, you are "Kepha" and on this "Kepha" I will build my Church - not "Churches".
In Matt. 16:21 - it is also important to note that it was only after Jesus established Peter as leader of the Church that He began to speak of His death and departure. This is because Jesus had now appointed His representative on earth.
Acts 1:20 - we see in the early Church that successors are immediately chosen for the apostles' offices. Just as the Church replaced Judas, it also replaced Peter with a successor after Peter's death.
John 21:15-17; Luke 22:31-32 - Jesus' creation of Peter's office as chief shepherd with the keys passed to Linus, Cletus, Clement I, all the way to our current Holy Father.
As for its infallible teachings, in Matt. 16:18, Jesus promises the gates of Hades would never prevail against the Church. This requires that the Church teach infallibly. If the Church did not have the gift of infallibility, the gates of Hades and error would prevail. Since the Catholic Church was the only Church that existed up until the Reformation, those who follow the Protestant reformers call Christ a liar by saying that Hades did prevail.
Even had Peter been selected to a post of authority over the apostles, which I believe not to have been the case, where is the scriptural authority for choosing a selection of a successor to Peter?