If a problem seems intractable, then, you may not be able to make headway by THINKING about it harder. That infamous burst of insight seldom seems to come while hunched over a desk or after that 10th straight hour in the lab - it comes "in a moment of distraction or else burst forth from the subconscious while we sleep"

It's not my experience. I usually find solutions in intuitively comprehended (visual) imagery that is purposefully developed (learned) to model the problem under study. The easiest example is geometry problems, other kinds of problems need translation. Of course, the solution itself isn't arrived at according to an understood algorithm, but your description makes the process less controllable than it seems to be. I believe the imagery is just another paintbrush handle, just as words are, but drawing with it is more of a craft than mysterious inspiration.