In these kinds of departments, I think the professor PI is understood as basically a management position -- they get the funding, they "hire" and manage the rig

In these kinds of departments, I think the professor PI is understood as basically a management position -- they get the funding, they "hire" and manage the right grad students and other staff who do the work -- and yeah, then they take credit for it. So they aren't "only" taking credit for others work, they are performing administrative and management functions. And if they do that job well, they will not stop getting students. Because it can be the only game in town (literally, in the "town" you are in), and switching labs is a set back (switching schools/departments even harder and more so).