I recently watched this Coursera course on learning how to learn and your post uses different words for some of the same things.

The course described what you call "shower-thoughts" as "diffuse mode" thinking, with an opposite called "focused mode" thinking and the brain only able to do one at a time. Focused mode uses ideas that are already clustered together to solve familiar problems while diffuse mode attempts to find useful connections between unclustered ideas to solve new problems in new ways. Not sure if these are the formal/correct terms from the literature that was behind the class, but if so it might be worth using them instead of making up our own jargon.

As for the class it definitely had some stuff that I still try to keep in mind, but it also had some things that I haven't quite figured out how to incorporate (chunking) or didn't find useful (some of the interviews). There is some overlap with what CFAR seems to be trying to teach. Overall I'd recommend taking a look if you have an hour or so per week over a month for it.