I have rather less mixed feelings on this topic than you, since I predated by only a few years the mandatory use of computers for written work production in secondary school. Of course computer use was forbidden: handwriting or nothing. If handwriting is slow and painful (I could manage 4wpm on a good day), why then you get marked down in absolutely everything and routinely castigated not just for bad handwriting but for excessive brevity and bad structure, because moving things around required writing them down again, so the first draft is the last draft. (Also a lot of teachers liked forcing us to take dictation, which meant I missed out 80% of what they were saying because I couldn't keep up, though since I couldn't read my own handwriting this was little loss).
When forced to an answer, the people imposing these rules said that good handwriting would always be essential, and that if your handwriting was not good nobody would ever read what you wrote, so it was reasonable to mark you down for every subject if your handwriting was poor. This is, you'll note, exactly the same argument as you're using here, in a slightly different domain, and it is plain that it is absolute nonsense. I never handwrite anything, nor have I in all my working life, and I have never suffered in the least for it. These days, people who handwrite are considered somewhere between eccentric and annoying, and certainly unnecessarily hard to read.
Now English spelling and grammar are harder to automate than typesetting, but it is likely that in a few years or a few decades we'll get there, and then your argument will seem as quaint and plainly flawed as my old teachers' do now.