Subconscious problem solving is definitely a thing as far as I'm concerned.

It's happened several times that I struggled with a bug for hours, then suddenly came up with a key insight during the commute home (or while taking a shower or whatnot), while not actively thinking about the problem. I can't explain this any other way than some kind of subconscious "brainstorming" taking place.

As a side note, this doesn't mean the hours of conciously struggling with the problem were a waste. I bet that this period of focus on the problem is what allows for the later insights to happen. Whether it's data gathering that alows the insights to happen, or even just giving importance to the problem by focusing on it. Most likely it's both.

Subconscious synthesis of seemingly unrelated strands of thought is the basis of assessing an event's meaning, for me. A day or two will pass after the event, and I notice the meaning evolves even without new information and without conscious thinking about the event.

For me, the subconscious is the wellspring of sudden insights.

an alternate hypothesis is that all thinking is unconscious with consciousness being the sum of many unconscious processes. So, while it's true as you say "that some kind of subconscious "brainstorming" taking place", that would just true all the time, and you only notice it when your conscious thoughts are of something else.

(n.b. historically speaking in the literature, "subconscious" was the word used to describe Jung's "woo-woo" ideas about a collective subconscious shared across populations, and unconscious was the word used for ideas about a single brain a la Freud which is what we are talking about here)