I think having solutions pop in your head is still compatible with my idea, if you consider it as a kind of intrusive thought, and also take survivorship bias into account. Intrusive thoughts usually have environmental triggers, which I argue is random enough. Moreover, often when the solution pops, you're not at the same place you were before: besides going to sleep, taking a walk or jog is also common advise (arguably sleeping usually also entails changing locale, so it's really a variation). IOW, you're "reseeding" your intrusive thought RNG.

Secondly, when the solution pops, my next thought is frequently "why didn't I think of this earlier?", meaning that the solution is quite obvious in hindsight, meaning had I just approached the problem slightly differently in the beginning, I wouldn't have to search very deep. So the solution didn't pop into my conscious thought because I've been mulling it over subconsciously, but rather the intrusive thought forced the problem back into foreground, but with a slight twist, and that quickly lead me to come up with the solution on the spot.

For intrusive thoughts where that didn't happen, I simply don't remember.

TBF, I admit I'm using a lot of weasel words, but I think it's possible to make my hypothesis more rigorous. Then you can design an experiment to falsify it, but I'm too lazy, and I'll leave that to real scientists. :-P