Alcoholism is called a "disease", but my understanding is that it's not the scientific meaning of disease, which is something that can be contracted and cured, but rather just a word to help people understand that there is a biological component that goes beyond someone's mere choices.
This biological component is genetic. So to ask whether a Paladin can "cure" genes, one is asking: Can genes be 'cured'?
Some would say that to answer this with "yes" leads to racism / sexist lines of thinking, since race and sex are other examples of genetic components. There is the similar ethical question often posed in science fiction about what happens when we try to make the "flawless human"? Does this make unmodified humans inferior?
Leaving ethics aside, and focusing only on the scientific / magical side of it, I don't think D&D has any precedence for clerical magic (particularly healing magic) to make permanent, genetic changes to an individual. However, wizards have polymorph, which could be viewed as a change to genes, and in older editions you could get "stuck" in the new form, making the change permanent; so it's not without precedence that magic, in general, can make lasting genetic changes. Flesh to stone I do not see as genetic, but more as conversion of matter (converting from one type to another, not a change that occurs solely within the DNA).
So, my conclusion is that for more than one reason, I would probably not bring this kind of thing into my campaign...
...well naughty word, now you've got me thinking though. It might be interesting to have a villain around this concept. They start as a traveling healer who claim to have found magic to "cure" the masses of their "imperfections". They start out seeming like a good person, or at worst a snake-oil salesman, but really they are creating a cult of magically "perfected" people, and once they have amassed enough devotees and come into power, their true side comes out.