But this would make it much harder to sight-sing typical melodies, eg one whose notes are all diatonic to G major.

Reason: a melody whose notes are all diatonic to G is almost indistinguishable from one whose notes are all diatonic to C in your system while in the current one they are quite distinct. Furthermore, your system would make it very easy to count chromatic distance between notes, but harder to count the diatonic distance (or the scale degree a note lies on).