At least as a place-holder answer: yes, primarily-undergrad colleges care about REUs, if only because it is the style, and because undergrads have come to expect it. Some undergrad places are surprisingly delusional about the "research" that their REU programs supposedly support. Never mind.
Indeed, it is unreasonable/implausible to imagine that any but the most elite undergrads would do any sort of serious/genuine research. Don't say that directly to anyone who apparently expects you to do exactly the thing that is absurd. The point is that what is truly expected is not genuine research. The true expectation, however couched, is that something palatable and new-to-the-kids be given to them to work on. After all, the real benefit of REU stuff is simply the non-classroom-setting, the collaborative nature, the non-adversarial relationship of mentors and students...
No, it is unlikely that even a fairly precocious undergrad will be able to engage with, or benefit from, looking at problems that make sense to a PhD'd person. It's possible, but to the extent that undergrad places care about REU-type-stuff, it's not really whether what the kids do is genuine (and they can't tell, in any case) but whether the kids are happy.
Problems that you imagine could be addressed by a sub-novice are what you want. Scarcely matters what "field". Right, typically perceived-as-nearly-prerequisiteless things work best, if only because of the perceptions of the kids. The practical point that REUs mostly occur just after peoples' junior years, so that the vast majority have taken no genuine math course at all, indicates the scope of the difficulty.
As a hint about how to not be excessively cynical/jaded, we can not that we do not encourage teenagers to act as though they were octagenerians, even tho', yes, eventually they will be. Not even as 40-yr-olds, etc. Various developmental stages are inevitable, and/but few people imagine how naive/ignorant they are. No sense in having that be the purported point!
One candidate starter-project is variant-on-standard-cliche, where the person develops the "obvious" parallel to completely hackneyed fun-thing. If they need the whole 10 weeks to figure it out, that's fine, and, if by chance/luck they run through it fast, it will have been educational, and may have provoked them a little. At that point, if it comes, you have to start thinking about a worthwhile project. NB, it's not "problem", but "project".
For myself: I did quite a number of REUs, with some quite-better-than-average students, some years ago, but it was frustrating... and I've not done any since. The frustration was primarily that they wouldn't genuinely listen to me, because their preconception was that admission to an REU vetted their own "genius"... So that, apparently, they surely were supposed to do whatever they thought best, seemingly because "old people" had no worthwhile advice to give, etc. The worst downside to this is the apparent belief that "the literature" really does explain things. I got tired of spending energy on people who so militantly ignored me. :)
But when one is younger, one can/should/must behave at a level of "idealism" that is insupportable later... since, otherwise, one may never have done so. :)