In theory this should be true, although the information asymmetries involved (especially for international students who go based on citations or notable papers

In theory this should be true, although the information asymmetries involved (especially for international students who go based on citations or notable papers or "reputation") mean that it can be hard to lose the flow of good students. Ultimately if they are a tenured prof and can keep winning grants, they can offer postdoc and postgrad contracts, and that is effectively the only internal currency that matters for a student that needs to find a postgrad position. With a limited number of chairs, it's hard for someone else to get a "rival" chair post and then compete internally, to give students the choice (which you'd get in a free market). If the institution has a good reputation, they'll likely have a flow of willing students regardless of how they act.