I would argue that no, Cure Disease is not intended to cure genetic disorders. I would however argue that magic should be able to cure genetic diseases. Perhaps a "Greater Cure Disease" is needed or "Cure Disability" (Restoration maybe?). Though I wouldn't place it higher than Reincarnate. Now, there's nothing saying that Reincarnate creates a healthy body, but neither does it say that it does not, that's some fluff room left up to the style of DMing and the theme of the campaign IMO, IE: In Ravenloft, Reincarnate and other forms of resurrection may not create healthy bodies.
Indeed, and your example of Reincarnation is kind of the crux of the question. Immunity to disease or effects that cure disease doesn't say it doesn't cure genetic disease, so should it?
It's DM fiat, I understand that. To be clear, I'm not suggesting there's a right or wrong answer to my original questions. Just curious to see how others approach the problem. How would you (generic) rule if the PCs ran across a kid with a cleft palate? Can the paladin cure it with Lay on Hands? Does Lesser Restoration's ability to remove the blinded condition remove cataracts acquired in old age, or congenital blindness?