If you reduce the number of staff, the people who are going to hurt first are the graduate researchers. I run my lab with a whole host of college and department staff who make all of our jobs easier. If you cut them, their jobs are going to fall to professors, and if they have to do more admin work, graduate teaching and research assistants are going to get more shit work, and also there's going to be fewer than them.

For instance we have a whole office that help us get our research funded. These people are "bureaucratic administrative overhead", but they make everyone's job easier by providing a centralized resource for this particular problem. Get rid of them an you can save millions of dollars in salaries, but you're going to lose more than that in lost contracts and professor/student productivity. This would mean students probably would get cut anyway, so they're making the smart move of supporting only the students they can, and not leaving anyone out to dry.