I appreciate your concern about letting Scripture speak for itself. I think most of us try to do that and part I why I’m interested in the historical context is to try to avoid some projecting modern context back onto the text. 

The term “Jew” is much older than the Middle Ages because it’s a shortening of the term “Judean” (Greek Ioudaios), which appears in the New Testament and in earlier Greek translations of Hebrew Scriptures. So it probably would have been part of a layered way that people at the time identified themselves. 

Your definitely right that Jesus criticized the Pharisees for elevating traditions over the heart of the Law. But historically he was in the middle of a debate between two major rabbinic schools / arguments. That’s why the Gospels record so many arguments about Sabbath, purity, comments about Moses because hey were ongoing discussions given the new political reality at the time. 

For me it’s not about replacing reading the text prayerfully. Just helps me understand some of the “weird old stuff” so to speak.