By the same token, the Patriarch of Antioch could claim that same title because he too is a Successor of St. Peter.
St. Peter went to Antioch First :)
You sound like you're saying I don't know Church history. What you fail to understand is the bishops of Antioch did not replace Peter as leader of the Church. Unless you claim that Evodius (bishop of Antioch for many years while Peter was still alive) was the leader of the Church at the same time as Peter.
St. Peter was leader of Antioch for a time. Then he left and lead the Christians in Rome. Because he died as Bishop of Rome, not Bishop of Antioch, Peter's successor is the Bishop of Rome.
Peter's primacy resided with Peter throughout his life. He did not leave it with Bishop Evodius in Antioch.