I've been thinking more about this... or, not really. One of the downsides to my particular mind-setup is that it takes a long time to retrieve things from long-term memory, but I did retrieve something interesting just now.

When I was younger, I think I did use system two moderately regularly. I do vaguely remember intentionally trying to 'figure things out' using non-synesthetic reasoning - before I realized that the synesthesia was both real and useful - and coming to conclusions. I very distinctly remember having a mindset more than once of "I made this decision, so this is what I'm going to do, whether it makes sense now or not". I also remember that I was unable to retain the logic behind those decisions, which made me very inflexible about them - I couldn't use new data to update my decision, because I didn't know how I'd come to the conclusion or how the new data should fit in. Using that system is demanding enough that it simply wasn't possible to re-do my logic every single time a potentially-relevant piece of data turned up, and in fact I couldn't remember enough of my reasoning to even figure out which pieces of data were likely to be relevant. The resulting single-mindedness is much less useful than the ability to actually be flexible about your actions, and after having that forcibly pointed out by reality a few times, I stopped using that method altogether.

There does seem to be a degree of epistemic hygiene necessary to switch entirely to using system one, though. I do remember, vaguely, that one problem I had when I first started using system one for actual problems was that I was fairly easy to persuade - it took a while to really get comfortable with the idea that someone could have an opinion that was well-formed and made sense but still not be something that I would 'have to' support or even take into consideration, for example. Essentially my own concepts of what I wanted were not strong enough to handle being challenged directly, at first. (I got better.)