Any meaningful public support program that actually keeps severely-mentally-ill visibly-homeless people from acting crazy in public will necessarily involve involuntarily incarcerating at least some of them, some of the time. A lot of very dysfunctional people genuinely do not want to cooperate with meaningful public support programs or are simply too out of it to meaningfully consent or not consent to participation in such.
That is going right back to false dichotomy. Something can be meaningful and not clean up the streets of all things uncomfortable.