[tripa]

Technically speaking, if they said it's G# it's G#.

G# minor is a much better use of the key signature system than Ab: 5 sharps versus 7 flats. In practical terms, that's a proxy for it being more common.

Your vision that minor scales are constructed from A minor is valid; thinking it's by adding flats exclusively is misguided.

I'm not going to go out on a limb and defend the fact that sharps-based minor scales could be more common than flat-based, as that's likely not the case. A much easier argument against your logic is that flats-based major scales are used all the time.

Any given key signature can be either major or minor, be it made of sharps or of flats. It can be seen as altering C major or A minor indeed, but the alteration is allowed to go either way.