Technically speaking that’s a Ab minor. New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s. Adding #’s are used for deriving new major scales. At least that’s how I understand it.
You can see this on the Wikipedia article on various minor[1] and major scales[2].
> New minor scales are constructed by modifying the A-minor scale (which contains the same flat notes as C-major) by adding Bs, not adding #’s. Adding #’s are used for deriving new major scales.
No, a major scale can have both flats and sharps and the same for minor scales. In fact major scales often start on a flat while minor scales often start on a sharp. Major scales use Db Eb F# Gb Ab Bb as the roots of the scales (rarely C# and Cb), plus the white keys; while minor uses C# D# Eb F# G# Bb (rarely Ab and A#), plus the white keys.
I think you're confusing two different ways of constructing the minor scales.
One way is to start with the A minor scale (which has no sharps or flats) and to go around the circle of fifths[1] adding sharps or flats. Whether you add sharps or flats depends on whether you're going clockwise or counter-clockwise: for example, D minor[2] (one step from A minor going counter-clockwise) has one flat, and E minor[3] (one step from A minor going clockwise) has one sharp.
Another way to construct a minor scale is to start with its parallel major[4] and add a flat to the 3rd, 6th, and 7th. But note that the result can still have sharps (like in the E example above, where E major has 4 sharps).
In any case, G# minor is definitely a key that is used[5].
[1] https://en.wikipedia.org/wiki/Circle_of_fifths
[2] https://en.wikipedia.org/wiki/D_minor
[3] https://en.wikipedia.org/wiki/E_minor
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.