I think the whole notion that the guy must be of help because he's genetically engineered is stupid. He might be more intelligent, but he did sleep for 300 years. If you brought Da Vinci into this time, he wouldn't be able to build a new aircraft carrier or aircrafts for the US Navy within a year.
This is sci fi, a genetic engineered human being can acheive whatever the writers want.
The fact that Khan is more than an ordinary human being is enough without going into all the boring details on how.
Those 'boring details' are probably what would have helped elevate this from fantasy to somewhat more credible SF. But I don't think they were going for that, or would have even recognized it as such.
AFAIK, the ONLY bit of Khan history that relates or supports his involvement with this uglyass ship is that he said he was something of an engineer in his day, back in SPACE SEED.