I am assuming some grant overheads also go to admin.

I get that, but a research university's prestige comes from the recognition for the research they do. Accepting fewer grad students means less research will be done and fewer papers will be published.

They could presumably cut admin staff to some extent, and pay grad students out of the tuition funds freed up. But why would we expect the bosses to fire their friends?

This is a complicated ecosystem, it's not that simple. Academic departments are not places where there's a lot of slack - positions are scarce, the competition for them is fierce, and the people who get them are notorious workaholics. Cutting admin means more work on professors, means less research output, means fewer grants funded, means fewer grad students supported. So you can cut students and get fewer students, or cut admin still get fewer students but also less research and funding as well.