I think I agree that my problem solving is connected with conscious thought, but the heavy lifting is mostly (or at least frequently) done by something that "I" am not aware of in detail.
When someone is explaining something complicated, pretty often, maybe not always, my (conscious) mind is pretty blank. I can say "yeah, I'm following you", but I feel like I'm not. Then when I start working on it, I feel like I am fumbling around for the keys to unlock some background processing that was happening in the meantime.
Also, when I am in a state where I am consciously writing something elaborate, and I feel connected to the complex concepts behind it, sometimes I get stuck in a blind alley. My context seems too narrow, and often I can get unstuck by just doing something unrelated to distract my conscious mind, like browsing news on my phone and then it's like a stuck process was terminated and I realize what I need to change on a higher level of abstraction.
It's possible I have some sort of inherent disability that I am compensating for by using a different part of my brain than normal, I suppose.
If I use a mechanical grabber aid to reach something, then it isn't figuring out how to do anything. But if I ask Wolfram Alpha the answer to a math problem, it isn't me doing it.