Very well written, except I should mention that "works of the law" is a translator bias, for the Greek only says "works of law" ergon nomos not "works of the law" ergon de nomos. That translators seem to add to the scripture is a pervasive problem when it comes to fine exegesis without the original language text.

For the record "ergon nomos" is a phrase that is tied to a Jewish colloquailism of the day that refers to community participation requirements. Much like, at a Baptist church they may have an unwritten code of dress for the men and and one for the women. "Ergon nomos" then is an identifying marker.

In context, "ergon nomos" or "works of law," is used by Paul to refer to the Jewish conversion ritual of the Pharisees, of which circumcision was just the first step. The other steps were to make an oath to do the Torah, perform a sacrifice, and then get immersed (yes baptism is a Jewish invention that is completely supported from the Torah).

Jewish conversion in Paul's day (as it still is seen today) was seen as a means to salvation for the Gentile, through a faith in Jewish identity, and not through law keeping (although that was the expected result). Ironically enough, the Jews believed in grace - that God had grace on them to make them born Jews, since they believed (and they still believe to this very day) that by being born Jewish one is saved (which Jesus clearly countered that by saying one must be "born again.") Thus for a Gentile to be saved, they reasoned he had to become a physical Jew. Obviously this salvation-by-Jewish-identity-alone is counter to the gospel of salvation-through-faith-in-Messiah-Jesus-alone, and Paul rightly argues this point beautifully in all his letters.

When you realize he was arguing against a false gospel of salvation by Jewish identity, instead of by keeping God's commandments like "love the Lord your God," it will help make a lot of his writings more clear. Since although Christendom has been theologically correct that keeping the Law doesn't save anyone, this unfortunately wasn't even the argument Paul was facing, since no one in his day believed such, and we therefore miss much of what Paul is teaching when we invent a strawman for him to rail against. The reason is because absolutely NO Jew, past, present, or future, ever believed that keeping the law saved anyone! Theirs was a belief in salvation by Jewish identity, not by law keeping. So what then is the law for the Jew according to them? To them, they believed it was the means to maintain their identity as Jews (rightly so), and thus saved (wrongly so, as this is the point Paul is actually arguing against).

So then every time you see the terms "circumcision" or "works of law" in Paul's writings, he's referring to the conversion ritual of the Pharisees that was done to make a Gentile into a Jew for purpose of salvation through Jewish identity alone (or in the case of the believing Pharisees that promoted it for the same reason, they coupled it as a necessary requirement to "secure" one's salvation through Jesus, which too is a bogus claim).

I hope that helps. I look forward to your responses.