I'd love to hear from John Carmack on this, it would be interesting to hear that he rarely has these 'shower' moments because he is always operating with great perspective.
I would expect the opposite instead. I don't know him personally and this is conjecture based on what I've read about him, but his being totally immersed in a particular subject, to the extent of going to a thinking retreat in a cabin, is the best way to reach those Eureka moments. I've written in the past about this, let me copy an old comment of mine: "Let me share a tip that might work for you as well then: some t…
Or maybe it's even simpler: conscious mind stuck on hard problem starts to work as broken record - you are repeating the same thoughts, reconsidering the same wrong solutions all over again. Focusing on something unrelated frees your uncounciousness to try something else.