I wonder where you get the idea that psychiatric detention is adequate health care, or that it helps patients. Regularly studies pop up pointing out that involuntary treatment in general does not lower the odds of suicide.

https://www.madinamerica.com/2019/06/involuntary-hospitaliza...

And before you ask, doing nothing does work, or at least it beats this option. Yes there are yet better options, actual interested attention from an actual human being, but literally leaving the person on the street works better than any mandatory treatment, and is a lot better than involuntary detention.

https://pubmed.ncbi.nlm.nih.gov/1023455/

And frankly, find a local Psychiatrist and ask them this question "Who is best of: a person committing suicide, or a person in involuntary detention NOT committing suicide for 6 months". They will illuminate what causes this problem.

So why are we doing this? Well, this gets done to those people to give Police an option to deal with people who've done nothing wrong but are judged to be "at risk" and generally being a nuisance, or the police judges they're "at risk" of doing serious damage by killing themselves, slowly getting worse but not actually having done anything wrong.

Lastly, some people are doing this because they see it as a form of euthanasia. They are ill, or have destroyed their lives (for example with medical bills) beyond repair and they just want out. It takes a long time for them to actually do something about this. Needless to say, any form of "treatment" for this will only (sometimes rapidly) make things worse. In fact most old people die not from old age but from starving or dehydrating themselves to death (this gets a lot easier once you're past 70 or so. Energy/calorie shortage causes cascading organ failure, which ironically feels good, and you fall asleep as your blood poisons itself. Obviously doctors do not make a habit out of telling next-of-kin this is what happened, nor do they try to prevent this by for example force-feeding them). But, there is sadly no age limit on this. For example, in one famous case a 15 year old kid relapsed into Cancer after a very intense and destructive cancer. She'd had chemo from 5 to 9 years old, which handicapped her phsysically and mentally, permanently. She caused a lot of damage killing herself by putting a certain person in a position where the police could indict her for murder, a hospital was involved, psychiatrists, she got people to cover up her earlier attempts at suicide, psychiatrists lied to her parents, her parents got divorced and partially remarried, youth services got involved because the parents did not agree on what to do ...

Note that this is independent of the actual health care system. The same problems manifest in Europe with public systems. The problem is that psychiatric help ... well, somewhere between doesn't help and actually destroys people.

Who pays for this is beside the point because mostly, of course, these people are homeless, very ill and/or (ex-)criminals. You can charge them all you want, but as the expression goes "you can't get blood from a stone".

The sad part is that parents and children can be held liable if a suicide (attempt) does damage, which then causes more problems. This is done to "protect" society against psychiatric patients and to redirect as much of the cost of totally rejecting people to those people and their families as possible.