Emacs doesn't just work out of the box. Nobody has time to fool around with Emacs and learn a bunch of non-standard shortcuts. Just to do what you're suggesting it would probably require installing an entire package manager within emacs, then using that package manager to install some packages, and installing whatever packages outside of emacs, and maybe you'd even have to put some lines in your .emacs file, and probably even more steps. There's a 100% chance there will be some issue along the way. Emacs is an archaic dinosaur. It has a high barrier to entry. Nobody uses it except computer programmers. Many graduate students can't even be bothered to install TeXMaker. A lot of them use Overleaf just because it's the path of least resistance.

TeXmacs has nothing to do with either Emacs or TeX. It's a standalone wysiwyg editor, like MS Word or OO Writer.

For me, it would just be more work without any benefit. For me, a pencil and paper is necessary. I can't think typing equations out. Part of it is probably just habit, but a good part of it is that pencil and paper support non-linear workflows effortlessly (e.g., drawing a box around a side calculation). In the end, I would end up writing the problem set up on paper, then transcribing it into TeXmacs or whatever for no real benefit. This isn't just theoretical. All through gradschool, I worked out research on paper before writing it up in latex for a paper or thesis section.

Well, this is exactly what it is for. Neither LaTeX nor Texmacs help out in the research phase. They are document preparation systems (although Texmacs could be used as a front end to some CAS's)

Why would you submit math homework that is handwritten? How would that save you time? How would that make it easier to grade?

Why would I type it up? That is a waste of time because, like I said, typing it up is just duplicate work. TA's are used to grading hand written work. I know because I did it.

Likely they don't know about it.