Whether you're a cradle catholic or not doesn't matter, the question is whether the Catholic claims actually trace back to the original church in Rome.
The original Church at Rome doesn't really mean much, because Peter didn't spend the majority of his life after Pentecost in Rome. That took a while. What matters is what the early Church taught. Whether Jesus gave the apostles the authority to teach, and whether he made Peter the head of the apostles, which, we believe, he did.
Matthew 28 "Go out into all the world teaching all that I have taught you." "Whoever hears you hears me." "Whatever you bind on earth will be bound in heaven, whatever you loose on earth will be loosed in heaven."
I've given reasons in this very thread that show they don't, as if there was going to be an authoritative claim of transmission one of the various crises the early church faced would have been the place to invoke it. But they didn't, not against the Montanists, not against the Marcionites, not against the Donatists, not against the Arians. The claim doesn't seem to have arisen and solidified until the 4th century into the mid 5th.
The Montanists, Marcionites, Donatists and Arians just show you exactly how the Bible cannot be the sole rule of faith. And yet the Fathers of the Church wrote against all those heresies, and labeled them heresies. Ignatius of Antioch, Justin Martyr, Iranaeus of Lyon, just to name a few.
Paul, in all of his letters, relies heavily on arguments from the Tanakh. He doesn't insist that because he is who he is his claim should be accepted but engages in arguments from the text to show that his position is correct.
Paul was writing primarily to the Jews he encountered in the various places, and used their scriptures to show his truth. Paul told the Corinthians in the first letter chapter 11 "If anyone has another practice, know that we have no other practice nor do the churches of God." Paul also said "The traditions I learned from the Lord I hand on to you."
Peter, Jude, James, and John also based their arguments in Torah/Prophets/Writings making direct allusions and arguments from them. Their disposition was not one of demanding based on who they were or an office they occupied.
Why? Because there were no NT writings yet, or very few?
No, but your argument rests on stripping a single passage from Matthew without context and insisting that it must mean what you say it means. Jesus states how His church is to carry on, who the authoritative teacher is to be, in John. The context of Matthew doesn't make sense for Jesus to be giving marching orders, in fact the context implies that Jesus is speaking of Peter's confession not Peter himself. Your discussion of Matthias is spurious at best, especially when we have Scripture like 1 Corinthians where Paul is chastising the members of that congregation for dividing the church based on the teachings of men and affirming that there is one head of the church, all others being members of a body. Yet you serve in a church headed by the Pope.
It actually makes perfect sense, in context. Jesus took them to Caesarea Philippi, completely out of the region, on a road trip miles from the Sea of Galilee. A retreat. He took them to the place where the Jordan River begins, where a monument to the god Pan was enshrined. And he asked them the question "Who do you say that I am?" Peter answered him, and was told that the Holy Spirit had guided him to that answer, and that the Holy Spirit would guide him in leading the Church after Jesus was gone. Of course, Peter was human and a sinner, Jesus knew that. He knew Peter would deny him. And afterward, Jesus asked Peter 3 times if Peter loved Him, and allowed Peter's forgiveness. And directed him to feed His sheep. He gave Peter the keys to the kingdom, making him His Minister after he was gone. Promised him the Holy Spirit to guide him. So it's a lot more than a single passage from Matthew...