I've thought on this a bit. I'm not sure directed thought on a hard problem is wasted, even if it's clear that an intuitive insight is required to solve it. It might be that long, directed thought on a problem primes the rest of the mind to gear its resources to exploring the problem space as well.

When I spend all day working on something intensely, it's not uncommon for me to have likewise intense dreams about it. Interestingly, the illogical mishmash that forms the dream is more jarring and disorienting than normal, perhaps because my brain is primed to be looking for incorrect ideas, since that's what I was probably doing all day.