I recall a very specific situation that occurred at a major metropolitan PD where I was taking 911 calls and doing radio dispatch. We got a disturbance call at an apartment. Upon contact with the party who was creating the disturbance, the responding officers immediately thought a crime had been committed. The party was a middle-aged male, wearing a woman's nighty, smeared from head to toe with blood and excrement. His apartment was knee-deep in trash, and he claimed he had killed his mother (and was wearing her nightie). We rolled a bunch of officers and they had to go through his apartment, literally shoveling trash up off the floor to try to find a body.
Turned out his mother had died recently of natural causes, he had lived with her all his life and had mental issues as well as a drug and alcohol problem (these things often go hand-in-hand). No crime had been committed except the noise disturbance. The blood and excrement was due to the man giving himself violent enemas with chocolate bars (no, I am not kidding).
Based on the damage he had done to himself, the officers decided to have him transported to the local hospital, where they got him stitched up and asked for a 72 hour psych hold. This was on a Saturday.
On Sunday morning, I took a call from one of his neighbors that he was raising cane again. We rolled a pair of officers out and he didn't answer his door. Got the landlord to let them in and found the guy unconscious on the floor, core zero. We rolled EMT and fire and they transported him, but he was DOA. Drug overdose was the official cause.
Why was he back in his apartment a day later, after he had been on a 72 hour hold? Because he was evaluated and found not to be a danger to himself or others. Clearly not correct, but I don't know how he comported himself in the hospital or what they did to make this determination. I just know the results.
I think there are a lot of reasons. Not least of which is that the severely mentally ill often do not think they are mentally ill, and will refuse any treatment and fight any involuntary confinement. They will not take meds, they won't go to therapy. And there is no legal means to compel them until and unless they cross certain thresholds.