That may require prohibitively large sample sizes, i.e. not be testable.

At least for certain populations the sample sizes should be pretty large. Also a smaller-than-desired sample size doesn't mean it's not testable, all it means is that your confidence in the outcome will be lower.

proxies generally fare even worse far from the mean than you'd expect from regression alone

Yes, I agree. The tails are a problem in general, estimation in the tails gets very fuzzy very quickly.