Why wasn't Paul one of the twelve apostles?
Paul was not one of the twelve because the twelve were apostles of Israel, chosen within Israel's borders. He was appointed the Apostle of the Gentiles and commissioned on Gentile ground. He had no connection with the twelve and could not fulfill the qualifications Peter laid out for Judas's replacement.
Les explains that Paul, formerly Saul, had absolutely no connection whatsoever with the twelve. The twelve were Apostles of Israel, chosen within the borders of Israel. Paul, by contrast, would be the Apostle of the Gentiles, so God chose and commissioned him on Gentile ground. He was not just a Jew but also a Gentile by citizenship, a Roman. Les also notes that some theologians thought Peter was remiss for letting Matthias take Judas's place instead of waiting for Paul, but that would never have worked; Paul would never have fit. In Acts chapter 1, Peter lays out the qualifications for Judas's replacement, and Saul of Tarsus could not fill that requirement.