Good points. I'm always amazed that the Roman Church used this Apostle as their standard bearer. But..... with their mis-interpretation of Matthew 16:18 I guess I see the logic.
Luke 22:24, [Also a dispute arose among them as to which was considered to be the greatest.] This happens some time after Matthew 16:18.....I guess the others had not received the word yet about Peter's primacy.
John 21:15-19, Jesus tells Peter three times to feed his sheep. Who are these sheep? Matthew 10:5-6, Jesus tells his disciples "do not go among the Gentiles but go rather to the Lost Sheep of Israel". Does not sound like he is telling them to go to Rome, does it? In fact there is no passage in scripture that tells of Peter being in or about Rome. In the book of Acts he is primarily hanging Jerusalem and vicinity. In 1 Peter 5:13 he alludes to the fact that he is in Babylon. The Romans try to tell you this is actually Rome but I try to take scripture at face value. If Peter wanted to have us believe he was in Rome he would have said Rome.
The Apostle Paul clarifies in Galatians 2:7 by stipulating that Peter's assignment was the circumcised and his, Pauls.....was the uncircumcised (Rome). In the same chapter Peter shows his lack of primacy by being afraid of those of the circumcision group, verse 12. If he had indeed been given the management of the early church do you think he would display this kind of fear?
Acts 15 is my favorite (Peter has no Primacy) chapter. At the council Peter does speak about the gentiles receiving the Holy Spirit but notice who is in charge. James, the brother of Jesus (The Apostle James was killed earlier in chapter 12). James says in verse 19...."It is my judgment, therefore".... This is the guy in charge....not Peter, and James is not even an Apostle. Remember, the lot fell to Matthias, Acts 1:26......not James.
In Romans 1:13 Paul says he wants to have a harvest among you as he had with other Gentiles (read uncircumcised). In Romans 15:20 Paul proves that Peter was no where around Rome because he says he would not preach upon another's foundation. Peter was never in Rome....it is false tradition.
Horse Hockey..He was not only in Rome, but was crucified there, and his remains are buried there. His tomb dates from around 100 AD. That doesn't mean he has primacy, but he was there as Bishop of Rome.