It is hard to overestimate the tendency to work the law and achievement back into our faith. It is utterly bewitching to turn away from free justification by faith to works of the law. But people do it. We are wired to establish our own goodness and we drift back easily. The point here seems to be: law and faith are mutually exclusive as means to establishing righteousness.
The purpose of the law was not to establish righteousness but to identify and restrain sin, to show the futility of works righteousness, and to bring us to Christ.
They linchpin for the whole argument is the quote from Deuteronomy 27:26 in verse 10. If you don't do the law you are cursed. Jesus solves for the curse. . . and you can't. Just because someone doesn't want there to be a curse doesn't mean there won't be one!
The other interesting argument is grammatical. Paul is so confident of the words of the Bible he argues the importance of the plural versus the singular (Gal. 3:16). Jesus is THE seed.