I really sympathize with your position. And be aware that this tension between (for want of better words) curiosity and performance will be even more critical if you want to make a living out of doing math past graduation. What it means in particular is that it is important for you to find the right balance between doing research and satisfying your curiosity, and then trying to educate yourself on the types of jobs that would be the best fit. (Yes, I know, as if you didn't have enough to do already, you're learning math research, but educating yourself on the job market is critical too.)
I will not be afraid to argue both sides of the issue: I think it is very important to continue to satisfy your curiosity. Eventually, you will graduate, and one of the first things you'll feel like doing is broaden your horizons. Going in a totally different field is probably not recommended, but apart from that, new ideas of research problems can come from unlikely sources, as noted in the comments.
So I would encourage you to stick to your universalism. After all, it's not for nothing that math folks, as a rule, get most excited about results that connect different disciplines. But I would encourage you to exert this curiosity mainly outside of your own field of research. This is a bit trickier to explain, especially since this will depend enormously on how technical your field is. You do have to read some in your field, but targeted reading works really well in research, and this might improve your productivity tremendously. Your long term goal is to become deeply knowledgeable in your field, of course. But in the meantime, trying to be broad in your field is a time sink that could work against you. Also, you may find better return on time invested after you've grappled with the subject without a net for a while.
By the way, you mention your REU: one of the things you may want to work soon after graduation is finding problems that are suitable for undergraduates (a big career plus these days, unlikely to change in the foreseeable future). These problems will not come from your thesis work, so a wider perspective will pay off here.