No, until the mental disorder is controlled. After all avenues of curing are exhausted, and it's still not under control, then they can be released, provided there's no significant risk that they'll harm others or themselves.
So you risk getting into "revolving door" admissions to hospital. A person with a psychosis becomes ill, rejects medication, is detained against their will in hospital and given medication against their will (sometimes by force) until they "become well" enough to be let out of hospital. At which point they're free to decide to stop taking the medication.
I'm not sure that this style of revolving door admission is better for the person than just letting them be homeless. (Although preventing people getting into that situation would be a good idea).
Surely not. The "revolving door" concept, as you call it, is the middle-ground between not helping at all, thus letting a mentally ill person reach trauma more severe, and permanently confining those with incurable mental disorders (but those which can be controlled, and the patient able to live healthily outside the ward), which is both wildly expensive and harder to justify overruling the patient's will and freedom.
If you want to provide MH treatment to people it's much better to start with children and young people and better drug and alcohol services, both of which would reduce the amount of homeless people, and also to provide better voluntary MH treatment for homeless people.
People aren't arguing with you because you want to provide MH treatment. People are arguing with you because you want to force that treatment on people against their will.