I think what I'm really saying is that if you build a system where mental health services are so shit and resourceless that people are shunted off elsewhere at every opportunity, to services that were never originally set up to deal with severe mental illness or psychiatric emergencies, then even if those services try their best to respond, people will want things from them that they could never provide.
Mental health services like mine offload responsibility onto universities because they have counselling services, but however much universities might want to help, or try to respond by hiring psychiatric nurses to deal with emergencies and seriously mentally ill students, they're not a psychiatric service and don't have those responsibilities or powers.
And of course it means that students who aren't severely mentally ill but want counselling because they're stressed or homesick or feeling a bit low or struggling with their workload will have a hard time getting an appointment, because of all the appointments taken up by people like me.
The solution to all this is properly-funded, properly-staffed mental health services with multiple routes in. I'm fairly sure that if a community psychiatrist had a suicidal student with bipolar disorder in front of them, and there was spare capacity for that student to see a clinical psychologist for bipolar disorder-specific psychological treatment, without depriving someone else of treatment, that psychiatrist wouldn't tell the student that they should just go sign up with the university counselling service.
Properly funding and staffing mental health services wouldn't necessarily directly help those students who kill themselves without accessing any services or who don't have a diagnosable mental illness, but it would mean mentally ill people being given the correct treatment by the correct people, and universities being freed up to provide more support to those struggling with life stuff and emotional troubles.