Plus your brain will crunch things in the background. How many time as a developer have you "solved" a problem you've been stuck on for hours or days while showering or traveling to work. Sometimes you can't see the forest for the trees.
Many of my more annoying problems have been solved that way.
I knew an old Vietnamese refugee who was an electrical engineer working on maintaining and repairing portable stage lighting ballasts. He used to tell me how he would dream about the circuits and debug them then after spending a week of just staring dumbly at the physical thing—not accomplishing anything while at work, then come in the next day and finish inside of an hour.
Maybe I think much differently than others, but sometimes on difficult problems I will think forever and do nothing. In the interim, people will think I'm an idiot ("don't you have work to do", "shouldn't you know this"). Then suddenly, whoosh, everything comes at once. I'll work for a comparatively short amount of time, code/work will just fly from my fingers, and whatever I'm working on will be excellent quality. One of my finest moments was in the process of one of these outbursts I was able to write a "one shot wonder" class that worked essentially exactly as expected without rewrites (no compilation errors or defects).
This is just kind of how I operate, and it really held me back in college (decent exam scores would bail out awful homework scores) though now it's nicer in a professional/adult environment where people give you a little more leeway, freedom, and respect. Though the drawback is that it ruins metrics like burndown charts and managements' expectation of a proportional work-to-output ratio.
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.
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.
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.