I highly doubt that senior administrators at colleges with single digit admissions rates are standing in their war rooms and pondering, “Gee, how can we make our college seem more desirable.”
Also, going back to your original post, which I’ve already expressed as being flawed, the 20-40 point difference in SAT scores between HYPMS and the-however-one-defines-as-the-next-tier is statistically insignificant IMO.
For both points, YMMV, of course.