I share your experience. But, I've recently read a book, The Mind is Flat by Nick Chater, a psychologist, wherein he argues that your brain does not crunch anything in the background, and only does what it focuses on.
However, the situation you outlined still occurs because you can get stuck in a certain paradigm (e.g. "this name that I can't remember definitely starts with 's'"), which you effectively reset, in addition to simply returning re-energised. Still a good argument for breaks and less strenuous working hours.
Yes, the brain doesn't crunch anything in the background, but your brain is a powerful association engine. Sometimes an outside stimulus will activate an association that will then associate with a solution you've been looking for. This is what happens in the shower: you allow your mind to wander, aka free-associate, and suddenly new possibilities unfold.
I call this the "attractor basin hypothesis" of problem solving- your mental state is in one attractor basin and you need to be in a different one; a break gives you another chance to enter the right one. If true, it may be possible to get the same benefit as taking a break and returning to the work by doing something that jolts you out of your current attractor basin. Maybe that just means asking the right question (c.f. oblique strategies).
Interesting idea. I'm currently learning German, which is rather similar to my native Dutch. Yet I'm learning most of it through English, because that's simply the language people tend to use for instruction material.
Every now and then when I'm doing English to German translation, I find myself puzzled and then have to remind myself "think about the Dutch word" only to then immediately understand/recall the German cognate. I'm sure someone talking to me in Dutch could have the same effect, and more broadly speaking, another 'attractor basin' may be activated by random chance, like the tired cliche of someone in a movie figuring out the 'case' as a result of their spouse or child mentioning something unrelated yet apparently analogous.
There are times when you're not focusing on anything specific, hence why you get ideas / breakthroughs in the shower, or when looking out of the window of a train / car or just mindlessly walking, because you're not focusing on anything and your System1 can solve stuff.
Though I feel that System1 can't work while System2 is focusing on other stuff like playing games or browsing social media so I don't think they're necessarily running in parallel.
However, the situation you outlined still occurs because you can get stuck in a certain paradigm (e.g. "this name that I can't remember definitely starts with 's'"), which you effectively reset, in addition to simply returning re-energised. Still a good argument for breaks and less strenuous working hours.