The Goenka schools provide a questionnaire to determine mental health and any medications the student will need. Each teacher is different in who they will let in, and all those schools are volunteer run. I can see that some teachers could be reckless with course admissions, but all teachers would have managed multiple courses before, and would know the difficulties for everyone if they let someone in who has a history of mental illness. It's not great situation for anyone if any student enters psychosis on a silent retreat, not the teacher/manager/other volunteers/other students.
Edit: Also it's tricky if your students are motivated enough to get in that they'll lie.
The Goenka centers do have a filtering questionnaire, but I think they tend to just get so incredibly much wrong on their handling of psychological problems. It's infuriating.
I will recommend that a friend does one, after giving some warnings, and making sure to talk with them afterwards, but I wouldn't blindly recommend anybody do one.
I imagine mental health issues are handled differently between schools and within them by the different teachers, but it's probably something that could be improved upon. There's probably also some merit to 'riding out the storm' for some people, but anyone with a family history of psychosis, or unmanageable psychiatric illness might want to think twice about attending, or at least be prepared to leave the course at first signs of trouble.
Or lying. I doubt the meditation center in the article updated the claims on their site after these events.