I wanted to make a variant of the Genasi Core stats but when I made the variant, I don't get an option to pick any of the Genasi subraces. Is there a way to fix this?

You would have to make it as a whole new base race instead of a variant, and then recreate the original subraces for it as well.

Variants are sort of like subraces, so they don't automatically get the subraces. You'll either have to create a new base race with the variant changes and recreate the subraces for it, or create a variant of each subrace of genasi with the base race changes.

Technically, variants are supposed to be “instead of” subraces.