But it is possible that other points on the scale where they don't, or are harmful.

Sure, there could be non-monotonicity.

If a school admits two populations, then the histogram of SATs of its students might look like a camel. But why should the graph of chance of admission?...Fifth, the other use of the regression model is that it lets them consider interactions, which do seem to say that there is not much interaction between SATs and other factors, that the marginal value of an SAT point does not depend on race, legacy status, or athlete status (except for the tiny <1000 category).

Imagine that Harvard lets in equal numbers of 'athletes' and 'nerds', the 2 groups are different populations with different means, and they do something like pick the top 10% in each group by score. Clearly there's going to be a bimodal histogram of SAT scores: you have a lump of athlete scores in the 1000s, say, and a lump of nerd scores in the 1500s. Sure. 2 equal populations, different means, of course you're going to see a bimodal.

Now imagine Harvard gets more 10x more nerd applicants than athletic applicants; since each group gets the same number of spots, a random nerd will have 1/10 the admission chance as an athlete. Poor nerds. But Harvard kept the admission procedure the same as before. So what happens when you look at admission probability if all you know is the SAT score? Well, if you look at the 1500s applicants, you'll notice that an awful lot of them aren't admitted; and if you look at the 1000s applicants, you'll notice that an awful lot of them getting in. Does Harvard hate SAT scores? No, of course not: we specified they were picking mostly the high scorers, and indeed, if we classify each applicant into nerd or athlete categories and then looked at admission rates by score, we'd see that yes, increasing SAT scores is always good: the nerd with a 1200 better apply to other colleges, and the athlete with 1400 might as well start learning how to yacht.

So even though in aggregate in our little model, high SAT scores look like a bad thing, for each group higher SAT scores are better.

Reminds me of Simpson's paradox.

But the coarseness of the buckets and the aggregating of schools does not allow me to draw much of a conclusion from this.

Yes, I don't think we could make a conclusive argument against the claim that SAT scores may not help at all levels, not without digging deep into all the papers running logistic regressions; but I regard that claim as pretty darn unlikely in the first place.

At the low end, I don't see why screening would break down. Why would someone with SAT<1000 apply to an elite school without really good reason? Yet lots of people apply with such low scores and don't get in.

They could be self-delusive, doing it to appease a delusive parent ('My Johnnie Yu must go to Harvard and become a doctor!'), gambling that a tiny chance of admission is worth the effort, doing it on a dare, expecting that legacies or other things are more helpful than they actually are...