In a generalized sense, I think having the students that are headed to HYPSMC etc locking down their choice early really really helps. Because they can and often do get into so many other schools, which exacerbates the problem we are discussing and trickles down to less competitive colleges. The exact best mechanism to promote this is an open question, of course.
One thing I noticed that does NOT help students give up their spots is that when they decline a school, it looks like they also are supposed to say where they are going. That encourages kids to hold onto the spot longer than they really need… Most kids with multiple acceptances probably figure things out in waves (semifinalists, etc). Of course, we would need more rolling acceptance admission policies to take advantage of kids giving up their spots earlier.
@DeepBlue86 @SlackerMomMD one thing that strikes me interesting about the private school limitation being discussed is that the target schools themselves probably also know about the policy, which might very well increase the admission rate for these students. (Either from the standpoint of increasing yield, or knowing that the students have thought very very careful which school is their top choice.)