The Jewish culture has different defintions of who is Jewish. For most, it is a matter of birth. If your mother is Jewish, you are Jewish. The DNA confirms it. I have no idea where this idea of Judah alone stems from. Perhaps it is mentioned in some Bible book. Of course, one can become Jewish by conversion. Many people in mixed marriages do so. You need to study extensively. Learning Hebrew is an important part. People say it is exhausting work.
They focus mostly on descent from mother's side. More secular Jews reject the emphasis on mother's line and state that the concentration camps included anyone perceived as Jewish. I believe Nazi race laws included descent from fathers. The Jewish culture is not as conversion as eager as Christians. When I first encountered Jewish college students, I thought they did not want to be friends. It took many discussions to understand that you can't apply Christian norms to people who are not Christian. They believe they have a special status with God but do not condemn nonJews.
What would be a failed Pharisee? I don't know that Paul failed. We can't know Paul. I suspect that he viewed himself as Jewish and that Jesus was the fulfillment of Jewish scripture. Paul started this trend. There were many Jewish Christians. The apostles worsihpped in the Temple after Jesus' death and resurrection. Dominic Crossan argued that Paul probaly did not start preaching to Gentiles per se. According to him, there were already Gentiles drawn to Judaism who would attend services. Perhaps they would convert if it were not for the requirement of circumcision. They were termed God-fearers. This group would likely me more receptive to Paul's theology than Gentiles in general.
Jews focus on mother's line because of the history of rape. If rape were widespread, the father would be unknown. Lineage through a mother was more concrete.
There are many views of Paul. I cannot keep them straight. It does seem strange that someone not an apostle who never met Jesus would claim apostleship. Paul claimed that his conversion presented a Jesus who was as alive and real as the Jesus the other apostles knew in the flesh. Paul proclaimed Jesus was present now in a believer's life. I don't believe Pauline scholars agree about what Paul believed. Of all the men we call apostles, Paul would agree most with Tammy. Christians believe in a link to the Old Testament and that Jesus fulfilled prophecy. Jewish people don't. Since I am not God and God does not talk with me, I have no idea what happened or what is correct. We don't know. I don't believe Paul was a failed Pharisee.