Your point about the "white keys" is well taken, which is why I mentioned moveable do solfege. The absolute pitch of "C" takes on a different name in every key, which respects its relationship within that key.

However, there's a reason that scales like the melodic minor are written with sharps (or "sharped flats"/naturals) ascending and flats/naturals descending. Sharps emphasize rising pitch and flats emphasize falling pitch, and can thus be bent slightly higher and lower, respectively. You would lose that with fixed chromatic note names.

(And as an aside, recognizing when a note in a scale has been modified from its normal position, and in which direction it was modified, is yet another reason to practice your scales!)