Genuine question: does the writing tool matter at all here if the exposition is clear and mathematically correct? I’ve seen great notes written in Word, LaTeX, and even slides—quality seems independent of format.

I would say it's not statistically independent. See https://scottaaronson.blog/?p=304 item #1. So we get to add another exception, which is fine.

both no in principle, and when you're used to reading LaTeX, word is ugly. It's a milder form of how if these notes were handwritten it wouldn't matter, but it would also be less appealing than them being typeset well.