There are a couple of obvious downsides like: What if it turns out the problem is too trivial? Move on to the next one. What if the interviewee has already encountered the problem before? No different than if the interviewer posed an already-solved problem.
It would be incumbent upon the interviewer to not reveal a solution if they come up with it first of course. And the evaluation of "how you approach and work through the problem" is less definite than whether an answer (brute-force or optimal) was achieved; but if approach is indeed the important thing, that needs to be evaluated regardless. I'm sure there are other downsides I'm not seeing off the top of my head.
I can't be the first person to come up with this strategy, nor try it in real interviews. Has anybody attempted this? Was it successful or not, and why?
I can't tell if this strategy that just popped into my head is of value :-).