> A simple improvement is to give each of the 12 common pitches within the octave a distinct position on the staff.
This doesn't work - the staff becomes too large, or the registral range it covers becomes way too small. The current staff has evolved to be readable without expressly counting the lines or spaces, via visual subitizing. If you add more lines, you lose this property and sight-reading becomes infeasible. Plus, diatonic pitch is actually meaningful in a musical sense, and it's useful to have a notation that references it. That's why we use for instance C, D, E, F, G and not <0>, <2>, <4>, <5>, <7>.
(This also describes the very real problems w/ Klavarskribo, BTW.)