That interpretation is a construct of the Reformation necessitated by the political need to mute the authority of the Church. Ironically it isnt even remotely scriptural.
The Catholic Church relies entirely on Scripture, and not a specious and forced interpretation, but on the a careful analysis of every word. Each word of scripture is as sacred as every crumb of the Eucharist. Lets first consider the words used to set the scene for the dialog in Matthew 16. Following the feeding of the 5,000 near Capernaum and the dialog in the Synagogue in which Jesus declared His body the Eucharist and real food Jesus took Peter and along with the other Apostles and Disciples up to Caesarea Philippi to reveal his papacy to him. The selection of the location is far from a coincidence. Caesarea Philippi is a 25 mile, two day hike from Capernaum through some pretty rough country.
Caesarea Philippi is located near the Golan Heights. The city, previously known as Panis, was built above a huge rock wall that was known in ancient times as the Rock of the Gods. It was a very important location militarily, and had been a place of temples and worship dating back thousands of years before the first century.
Physically, that rock stood between a city of temples and the gates of hell. At the base of the cliff is a massive grotto or flooded cave that at the time was a natural cistern that the Greeks and their predecessors believed was a gate to the underworld. Although now collapsed by seismic activity and dry the Romans tried to measure the depth of the cistern and gave up when they passed 800 feet without finding the bottom. It is interesting to note that the ancients used to perform sacrifices in this cave and would see the emergence of blood from the sacrifices in the local springs that form the headwaters of the Jordan River as a sign.
It was before this wall, topped by the temple to Pan, the Greek God of chaos and confusion, that the conversation took place. In the presence of the massive rock that was a foundation to a holy city Jesus told Peter that he was a smaller rock, and that upon it, like the larger rock upon which was built the city before them, He would build His Church and that it would stand forever against the gates of hell. A side note is that all of this arguing about the relevance of Petros versus Petra is foolish in the context of that location. Jesus often used wordplay and humor to convey His message.
Jesus began the dialog by asking His Apostles and Disciples who the people said He was. He was demonstrating His rejection of a democratic, self-interpreting Church. The response to the question was varied. Some said John the Baptist, some said Elijah, some said one of the prophets. All were wrong.
Jesus then asked the Apostles collectively who they thought He was and there was silence and He then rejected an oligarchical or elite governance. Then, without consulting the other Apostles, Peter stepped forward and declared that Jesus was indeed the Son of the Living God, which was the correct answer. Jesus acknowledged this and declared that Peter could only have learned this from God, designating that like the He had throughout the history of His people God had chosen one person to be the spiritual leader to carry on after Jesus. And there, before the rock that stood between the ancient temple city of Caesarea Philippi and the cave grotto that was known to the people on the region as the gates or jaws of hell declared, it was upon Peter that he would found his Church, His Ekklesia, His Qahal.
Peace be with you
“Jesus then asked the Apostles collectively who they thought He was and there was silence and He then rejected an oligarchical or elite governance. Then, without consulting the other Apostles, Peter stepped forward and declared that Jesus was indeed the Son of the Living God, which was the correct answer.”
There is no evidence that there was hesitation among the Apostles, or even anything that singles out Peter beyond the fact that he answered. This is also the same Peter who answered for the Apostles that they would not deny Him, yet he denied Him thrice. While Peter is certainly a worthy Saint (and the scripture teaches that we are ALL Saints, a Holy Priesthood and “Lively stones”), there is no evidence that he was superior to the other Apostles in the actual Gospels. If anything, his humanness is confirmed, as well as his superhuman change from materialist and haughty temper to warrior for Christ in Acts.
Your interpretation reads things that are utterly unnatural to the entire body of scripture which makes it clear that Christ is the Rock, and that we are “lively stones” in that house which has Christ as the Chief Cornerstone.