Private universities that provide from $40k to 50k financial aid if accepted?

I understand that a lot of top universities said they would provide/meet 100% need of the students. However, some of my seniors who got into big uni like Cornell, UPenn, Vandy,… all wrote that they could contribute around $20k and ended up having to pay up to $26k.

Besides need-blind schools for international students, what are the universities that are need-based but still willing to provide a lot of financial aid?

Start here:

https://en.m.wikipedia.org/wiki/Need-blind_admission#U.S._institutions_that_are_need-blind_for_U.S._applicants_and_meet_full_demonstrated_need