I agree that intonation matters a lot; I guess I'm thinking about how you'd make a notation that better conveys the information already there in the current scores, which is 100% equal tempered. I actually like the idea of modifiers for microtones, and presumably any of that stuff would work just as well on a 12-tone staff.
Also, with a 12-tone staff plus shape notes, you'd get a little extra information for just intonation because you can tell for sure what key was intended for a given note.
Even in the key of C major, this is a problem in just intonation. Say you want to play a G major chord, so it's made up of G, B, and D (3/2, 15/8, and 9/8). Later in the song you want to play a D minor, so you play D, F, and A (9/8, 4/3, and 5/3). That doesn't sound right, though. It turns out that the D you want is actually 10/9, which is just a bit flatter than 9/8. In standard notation, you can't distinguish.
It's possible to get around this by adding non-standard modifiers to notes aside from the usual ones (sharp/flat/natural), but unmodified standard notation misleads people into thinking that those two notes are the same. Which is another example of your main point, that "standard notation gives a profoundly misleading picture of how music and harmony really work".