I appreciate the good faith attempt to rationalize the idea, but it's fundamentally flawed.
We (as in Western Civilization living in post-Enlightenment liberalism) can't purport to respect human dignity and self-determination and also criminalize homelessness. And that's what jail is, criminalization. There is no "non-criminal" jail. In the United States, you can't be criminalized without due process, because fundamentally, a basic western human right is the right to personal dignity and autonomy.
Criminal recidivism is a problem. Some homeless or mentally ill people will be recidivist criminals. They should be punished for their crime after being afforded due process and any defenses thereto (e.g. lack of mental capacity). There are ways to handle criminal recidivism (in CA, famously, there's the "three-strikes" rule against multi-time felons). Apart from hardcore criminal-justice reformers, nobody disagrees with the idea of punishing crime according to the nature of the crime and increasing punishment for recidivism.
Being homeless is not a crime. Nor is it a crime to have mental illness. There are ample failures (drug war, lack of safety net, unwillingness to regulate predatory housing/investment, lack of accessible mental healthcare, social stigma, etc.) that cause homelessness/mental illness. All of which are indictments against society at large. It is anathema to a free, liberal culture to cause the problem and then punish it criminally.
To say that society only intervenes when a crime is committed is reductionist and wrong. There are plenty of civil remedies that restrain conduct (e.g. an "injunction"). More broadly, the government restrains conduct all the time for non-criminal matters (taxation, zoning permits, most financial or environmental regulation, intellectual property laws). Different efforts to restrain activity have different methods of enforcement. Including involuntary civil commitment. But for the reasons mentioned above regarding due process and personal autonomy, only crime is enforced by jail time.
We cannot claim to be a free, law-abiding, liberal democracy rooted in Western ideals and at the same time carte blanche suggest that "jail" is a categorical solution to homelessness and mental illness, as the parent commenter suggested. The type of thinking that the parent commenter proposed is anti-democratic, anti-western, and candidly, fascist. There are plenty of places people can move to if they want to live in that type of society, but none of them claim to be Western in design.