For grad students, yes, the money largely comes from grants (more than 50% of my funding in grad school was from grants from my advisor). The university doesn't actually take a chunk of the grant, they get an additional "overhead". This used to be 60%, now with the Trump administration its been cut to between 15% and 30% depending on the federal agency. For example, if the overhead was 30% and I get a $1000 grant from the NSF, the university will get an additional $300, but I still get the full $1000 (actually the university has full control of the $1000, but I can ask them nicely to use it).

I am more concerned with the research institution's effect on undergrads rather then grad students. Grad students have their own unique ways of being exploited.