Independence in scientific direction? Yes. Independence in personnel management? No.
Independence in scientific direction? Yes. Independence in personnel management? No.
There's a fundamental organizational change that needs to happen within universities...
Also, funding comes straight from the professor. If your advisor decides they don't have funding, you're out of luck and working on the side, unless your department is nice enough to give you a teaching assistant job.
Is this possible?
A heavy, bureaucratic firing process protects underlings (somewhat) from abusive bosses, but at a cost of protecting incompetence. You'll see this in any big corporation, and a great many academic jobs as well.
The whole idea of tenure in academia is to protect academics from retaliation so they can speak freely. But their lab assistants and secretaries and such get no such protection.
I think it is quite rare to see grad students fired. Typically there are two strategies to get rid of a graduate student:
1) Just graduate them (either with a masters or with a undeserved PhD)
2) Make their lives miserable so they quit.
I don't think switching the source of funding for graduate to the department really prevents either of these from happening.
i) Admit tons of grad students, treat them like shit and just assume many will drop out/quit so you don't have to do any long term planning on resources/admissions.
ii) Plan projects and resources, vet candidates and treat grad students well. Attrition will be lower so this does allow for longer term planning. In addition, here the institution and department are involved in the planning/resource allocation and presumably personnel issues as well. In general culture will be professional.
IMO i) gives signals that harassment and other mistreatment is tacitly allowed. With ii) the grad students are valued and signal is given that this should be a professional environment. Another generalization, but I have observed American universities, especially engineering departments are close to i), European universities (some countries more than others) are closer to ii).
Not to say switching funding to the department is a solution, but a professor's power over the funding is definitely a significant problem and source of great risk for students who are being sexually harassed by their professors.