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.
Yeah in my experience professors are pretty good about giving credit (the student is the first author on the paper, does the conference presentations, etc).
The comment above is right about the general structure (professors pitch, students deliver), though. And perhaps right about claiming credit in some fields, although not mine.
I'd agree it very much depends on the individuals. I have worked with people all across the spectrum. My advisor was absolutely in the "good" camp - we'd openly describe call academia the "world's biggest hidden pyramid scheme". He led a small, tightly knit group, and the group followed the two pizzas rule.
I've also dealt with people who tried to come across like this, but were the absolute opposite - they'd grown a pretty large research group to get their chair, but had a reputation of over-promising and under delivering. They'd promise things without the technical knowledge to understand what they were saying they'd do, then causing morale problems among their team when people found out what had been promised. They absolutely didn't follow the two pizzas rule - you could barely find their whole group, let alone fit them in a room. This prof would try to give the impression they wanted others to grow and succeed, but ultimately it was all about them, and preserving their carefully crafted hierarchy. They didn't like to hear "no", and couldn't bear the thought that they were over stretched and not delivering, so everyone would tell them things were fine, as they under-delivered, exacerbating the situation...
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).
