Is there any place where such incidents be reported? Preferably outside of the university system, since universities would naturally have a conflict of interest.
In my opinion, there is no way to achieve any such fair venue to record mishandling of students. The reason are as follows:
This is not unique at all to universities; mishandling can occur in any work environment.
There is no way to assure the fairness of the recorded instances. Since human civilization has invented and developed the concept of due process precisely for handling all such conflicts between parties, and the states/countries are the authorities that invest money and build the infrastructure for maintaining all aspects of fairness and due process (the legal system), there is basically no way (at this point in time) to replace it with websites purporting to "expose" mishandling of students or employees by professors and employers.
Many universities have a standard process to handle these cases. I agree with you that one of the considerations students should make when picking a graduate school is the existence of a strong student support system, and options for recourse against bad advisor behavior. Of course, no one ever thinks that this would happen to them, so often enough this is not considered.
Are there websites that "expose" bad professors? Not to my knowledge. However, it is easy enough to find out before applying. I strongly suggest that students talk with existing students of a PI about their advising style, their interaction with the PI, and how much supervision they actually get. If a professor is unwilling to let you speak with their students, then this is a huge red flag.
As @Dilworth says, the main issue with the existence of such a website is that it is impossible to ensure that accusations on the website are not frivolous or unfounded. An accusation of misconduct towards one's advisees is a serious matter, and should be treated as such. Often enough, there's no one side that's obviously wrong: students sometimes "disappear" for long stretches of time, ignore the PI's advice on choice of research directions, or are simply not suited for grad school. The tendency of some students is to blame their PI for their failures. Ultimately, the advisor-advisee relationship is very personal.
I have seen instances where an amazing advisor with an excellent track record of successfully graduating students had to drop a certain student because they weren't a good fit, hurting that student's future prospects. Should that professor be held "accountable" for this?
Similarly, I know of advisors who have very abrasive and unpleasant advising styles, but were able to graduate several stellar students who are now leaders in my field. Are they on your website?
There are several legal issues with such a website. Posting something so damaging on a public forum may have legal implications for those posting it, at least in the United States. You may have lawyers representing the university, the faculty union (if the professor belongs to one), or the professor coming after you for posting damaging accusations online.
Finally, there is the question of incentives. Suppose that I see that Professor X's former student is now applying for a position in my lab/company, and that they posted a scathing review of Professor X's behavior on your website. Talking with people close to the matter, I get mixed signals - some side with the student, some with Professor X. Would I want to hire this student? Probably not. You might argue that this can be anonymous, but it's really not very difficult to figure out who the poster is - after all, there are only so many people in a research lab, and only so many of them who drop out/are not doing well. Thus, it may not even be in the students' best interest to have these accusations posted online.
To conclude, there is indeed a power imbalance between advisors and students, which both sides should be aware of and treat with great respect. As a student, you should ensure that you know your rights, and that your university has a support system for you to protect them. I don't think that your proposed solution is the best way to handle it.