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.)
A currently existing social norm basically says that everyone has the right to an opinion on anything, no matter how little they happen to know about the subject.
But what if we had a social norm saying that by default, people do not have the right to an opinion on anything? To earn such a right, they ought to have familiarized themselves on the topic. The familiarization wouldn't necessarily have to be anything very deep, but on the topic of e.g. controversial political issues, they'd have to have read at least a few books' worth of material discussing the question (preferrably material from both sides of the political fence). In scientific questions where one needed more advanced knowledge, you ought to at least have studied the field somewhat. Extensive personal experience on a subject would also be a way to become qualified, even if you hadn't studied the issue academically.
The purpose of this would be to enforce epistemic hygiene. Conversations on things such as public policy are frequently overwhelmed by loud declarations of opinion from people who, quite honestly, don't know anything on the subject they have a strong opinion on. If we had in place a social norm demanding an adequate amount of background knowledge on the topic before anyone voiced an opinion they expected to be taken seriously, the signal/noise ratio might be somewhat improved. This kind of a social norm does seem to already be somewhat in place in many scientific communities, but it'd do good to spread it to the general public.
At the same time, there are several caveats. As I am myself a strong advocate on freedom of speech, I find it important to note that this must remain a *social* norm, not a government-advocated one or anything that is in any way codified into law. Also, the standards must not be set *too* high - even amateurs should be able to engage in the conversation, provided that they know at least the basics. Likewise, one must be careful that the principle isn't abused, with "you don't have a right to have an opinion on this" being a generic argument used to dismiss any opposing claims.