To the rest of your scenario, no one says they're "fine". Just that the person has the right to choose no medical treatment. You also have that right. If a doctor says you will certainly die without treatment, you can say no. The doctor doesn't write in the note that you are fine, just that they offered and you did not want it. It's not a "game", it is the ethical and legal requirements of the role.
You assume your proposed intervention will "help", but what is your basis for that claim? What signals should they be looking at to guide the course of treatment? For doctors, the answer can be vital signs, tests, but those don't help here. Usually, they're looking at patient input.
> If the people advocating for the homeless and mentally ill have a better solution than leaving them on the streets to rot, then I would like ot hear it.
Properly funded, voluntary community treatment. It is possible to design services such that people want them; I've seen it (yes, even people experiencing psychosis). You have to train providers in engagement skills and actually empower the person receiving services. Traumatizing them with regards to health care is not going to help.
Fixing it requires and robust social safety nets at the federal level. A municipality cannot do it alone.