[hammock]

Yes. Equal temperament does not explain why there is Ab and G#. It merely explains why they have not always been harmonically equivalent.

You, and the top SO answer, give the correct reason why. And it's not just for writing, but reading also.

The diatonic scale, combined with the staff of lines and spaces, necessitates both flats and sharps.

[seertaak]

> The diatonic scale, combined with the staff of lines and spaces, necessitates both flats and sharps.

Careful -- you're in danger begging the question. See my remark regarding mathematical summation. Intervals are nothing other than a pitch distance. Note names are the absolute value of a pitch. I'm not at all convinced that it's not possible to design a notational system that does away with the sharps/flats and yet retains the compactness optimality w.o. the ionic scale.

[hammock]

The critical thing here is that the diatonic scale (of which ionian is one) is whole, whole, half, whole, whole, whole, half. There is no symmetry and the pitch distances are not consistent. How would your proposed novel notation system handle that?