They were both part of the Church that Christ founded. However, if you will recall from the Scripture, Jesus gave the Keys to the Kingdom of heaven and earth to PETER. These were not keys to a new Honda. These were keys that gave Peter authority. Jesus even changed Peter’s name from Simon to Peter, which means Rock. You are wrong in your interpretation of that passage in Scripture. I would challenge you to find a single early Christian who thought the rock in Matthew 16:18 was “a revelation” rather than Peter. The number of times God changed names of people is small-- Abram to Abraham, Sarai to Sarah, Jacob to Israel, but they always meant something huge was happening. This is so often discounted by protestants because they do not read Scripture in the context of the time it was written and they often do not see the parallel between the OT and the NT. Peter was the one who went out and spoke to the crowds immediately after Pentecost. Peter was the one who is clearly making the authoritative decision in Acts 15 at the Council of Jerusalem. He spoke and all were silent. Did Jesus give Paul keys? No. Was Paul in the upper room at Pentecost? No. Was Paul preaching the Gospel in the first days of the Church, no, he was busy persecuting the Church-- which Jesus even said was persecuting HIM. Paul came in after the Church was established. That is why it is not really strange that Paul was not the first Pope. However, I would never dream of trying to undervalue Paul’s work-- his service to the Lord and to His Church was truly amazing.
As to the root of your question, the Catholic Church existed before the New Testament. Jesus gave His authority to the Church which, as Scripture attests, is the pillar and foundation of truth (1 Tim 3:15). We are instructed by Paul to hold fast to the traditions we have been taught either orally or by letter. Catholics and Orthodox (the other ancient Church which is very similar to Catholic) actually follow that instruction. Everything in Scripture is completely compatible with Church teaching. But you have to understand the context in which the Scripture was written and that is where many modern day non-Catholic Christians fail. I always find it amusing that they fully believe that God guided ordinary men into writing inerrant Scripture, but they do not believe that God could guide a shepherd into infallible teaching. Be that as it may, there are many things that Jesus did that were not written in the Scriptures, as the Scripture says. Who do you think hung on to all those things? Do you think that a non-denom Church founded in 1984 has that information?
Some historical issues to consider. The first time you see the name Catholic Church is in 107 ad! (St. Ignatius letter to Smyrna). Consider that when archeologists dig up Church’s from the earliest centuries they always have one thing in common-- an alter for offering. Most non-Catholic churches don’t even have alters anymore. Consider the Didache, and the early Church Fathers. They may not be inerrant but they do say what people believed and were doing at the time.
Here is a website that is full of Scriptures which show how they actually support Catholic teaching. However, the Church is not a religion based on the Bible although we love the Sacred Scripture. How could it be when the Church was here first?
PS… As to the sprinkling of water in baptism, it arose from necessity. First of all, bringing crowds of three thousand to the Jordan would not have been easy. Second, and I believe of more importance, was the fact that early Christians were persecuted and stayed out of sight. As to the
Mass, we see that initially in the Last Supper, but many foreshadowing Scriptures as the link will show. See also in Acts 2:42 “And they devoted themselves to the apostles’ teaching and the fellowship, to the breaking of bread and the prayers.”