Harvard did. They went to court multiple times, and won multiple times.
The problem is that the administration doesn't care whether the schools did anything wrong. The investigation is the point. The harassment is the point.
It's not a schools-caving problem. It's a SCOTUS problem. What are schools supposed to do when they see that SCOTUS has stayed like 15 injunctions in a row involving Trump overstepping his authority? Why litigate to get an injunction if the court is just going to stay it?
This is why I say SCOTUS is no longer a court and needs to be institutionally destroyed. Their goal is to decrease litigation against the administration's rampant illegality, so they don't have to confront the hellscape they created. They just stay judgments without thinking, and go home to their wives with flags and lucrative jobs placing clerks with law firms. Meanwhile, in the real world, what to do?