It's worth clarifying, because you absolutely can sign away rights in a contract. You can even sign away constitutionally protected rights, like free speech. This regularly happens vis-a-vis NDA's.

I bring this up because I am (morbidly) interested in the technicalities. Like, could it just be an unenforceable provision to an otherwise valid contract?

> You can even sign away constitutionally protected rights, like free speech.

The text of the first amendment only asserts that "Congress shall make no law [...] abridging the freedom of speech", so signing away your ability to speak isn't in conflict with the Constitution.