Why do anti-mold products avoid stating it kills mold?
Products that remove mold stains are generally cleaning products, not designed to deal with the mold spores. Killing the spores, in a meaningful sense, involves specific products to kill spores (like this one).

The reason why you need a spore killer specifically, is that cleaning surfaces (like with bleach) can kill the mold on the surface, however it does not prevent mold regrowth. So you can bleach a surface and still have mold come back and be just as bad as it was before.