Petros v. Petra
The initial problem with this analysis is that it looks at Greek, not Hebrew or Aramaic. While the Bible was written in Greek, Jesus spoke Aramaic.
It does raise an interesting point that Peter denied Jesus and so the Gates of Hell prevailed against Peter (sort of, kind of, not really). Peter sinned by denying Jesus, but then became one of the people proclaiming Jesus.
However, Jesus says that he WILL build his Church on a rock, not that he IS building his Church on a rock. This indicates that at some future point the Church will be built, not that it is being built at the present time. Examining this passage, Ephesians 2:20, Paul says that the household of God is built on the foundation of the apostles and prophets. Jesus is the cornerstone. If you look up a cornerstone, it is the first stone laid and all other stones are set in reference to it.
There is a simple explanation why Matthew used the masculine form when referring to Peter; Peter is a man not a woman, ergo, the masculine is used. However, there is another realization here. First the masculine is used and then the feminine. The masculine means movable rock; the feminine means immovable rock. This refers to Peter’s faith not being solid at the present time (movable) and that it will be solid (immovable) in the future.
Every single translation I can find of Acts 4:10-11 follows it by saying Jesus is the cornerstone. See earlier comment about cornerstones.
If you examine 1 Cor 4:15, while it is through the Gospel that they are begotten, Paul does beget them (through the Gospel). They (through the Gospel) are the children of Paul.
We have in Luke 9:46, that the disciples quarrel. Later in Luke 11, Jesus gives the Lord’s prayer. In Luke 13:18, Jesus repeats the parable of the Mustard seed. In Matthew, the Lord’s Prayer is given in 6:9. The parable of the Mustard Seed is in 13:31. Peter is appointed Chief Apostle in 15:18. The quarrel happens BEFORE the appointment.
As for there being no title of Pope in the Bible. Okay. A simple explanation is that while the office existed in some form or another, it had not yet attained the title Pope. Also the true title is Bishop of Rome. During Acts 2, Peter was not yet a Bishop, let alone Bishop of Rome, ergo, he could not hold the title Pope. That does not mean he was not the leader of the Apostles. Which is seen in Acts 2, when Peter is the one to address the crowd. In Acts 4, it is Peter, not John, that is filled with the Holy Spirit. Skipping ahead to Acts 10, the repealing of Kosher Laws is proclaimed to Peter, not one of the other Apostles. If anything this also confirms the notion of Papal Infallibility as an infallible proclamation (no need to keep Kosher) was made to Peter. Fast forward to Acts 15, after a lot of debate, Peter stands up states that it is by his mouth the gentiles should hear the Word of the Lord.
At the end of the Gospel according to John, Jesus specifically calls out Peter and gives him commands. The Church maintains that the Pope is to be the most humble of all priests. This is not to say they always act like that. In Matthew and Mark, when Jesus goes to pray in Gethsemane, he takes Peter, James and John deeper into the Garden. When he comes back, they are sleeping. He specifically calls out Peter rather than the other two; he is holding Peter to higher account.
Bottom line: you trust Rome and the Pope. You elevate them over God and His word.
I’m done here, but I will pray of you.