On the sharp end, E# minor and A# minor are rarer than G# major and C# major respectively, as mentioned in the question. Generally, major keys have closely related minor keys with more flats/fewer sharps, while minor keys have closely related major keys with more sharps/fewer flats, so extremely sharp key spellings occur more often as major keys, while extremely flat key spellings occur more often as minor keys. So if a piece is primarily in C# minor and modulates to its dominant major, it’s often more readable to write the section in G# major than to switch between sharp spellings and flat spellings within the piece.

These keys do appear often in practice (since their enharmonic equivalents, Db minor and Ab minor, have more flats than these have sharps). But G# major is closely related to several minor keys which have fewer sharps in their key signature - most closely, C# minor and G# minor. In isolation, G# major is just as impractical: for the primary key of a work, one would almost always write it as Ab major instead. But this provokes the followup question: if E# minor so rare as to be “purely theoretical”, why is G# major (with the same key signature) less rare in practice? As other answers point out, E# minor exists in theory, but is in used vanishingly rarely (if ever) in practice.

G flat sharp major scale