I occasionally ran into exam questions that were "discuss the motivation behind making this maneuver instead of some other maneuver in this proof we discussed in class," which is a sort of meta-showing-of-work.

Is it? Proof to me means I can say "here's the answer" without saying how I found it, maybe I just tried a few random things, and just prove that it's correct.

For instance, for a polynomial I might show that my answer solves the equation, and that I have as many solutions as the degree.

Maybe that goes for "show your work" too? As I didn't study in the US, I don't know what a teacher would expect when saying that particular phrase.

Yes, heuristics is important and underrated in math education, especially in the US, where they place excessive faith in rote learning.

Proofs are sometimes a little unsatisfactory, the theorem is proved satisfactorily, but all the traces how you got to the proof are covered.

if you're asking a proof question, it will be something of the form "prove that there are as many solutions as the degree for all polynomials" rather than a specific polynomial.

if you skip writing down each step in the proof, aka your work, you haven't completed the proof.