Describing prison as "housing at taxpayer expense" is... one way of putting it.
I think they should be given a choice: go to jail or get a job and 2 months of state funded housing. If you go back on the streets because you didn't want to work, then police will pursue every public nuisance, public urination, shoplifting, etc. charge and send you to jail. If you're mentally ill you're forcibly committed. If you're a drug addict, you're forced to go to rehab or go to jail.
Otherwise these people will never learn to be decent members of society (you know, people that don't steal and shit on the streets - the bar is on the floor).
A lot of these people don't want to go into shelters because those shelters have no drug use/no alcohol policies. But I say the state should not be supporting their drug habits with taxpayer dollars.
While the parent comment describes a scenario where the person is punished for an actual crime, and they get (in addition to the default option of serving time in prison) an option to remediate it by getting housing assistance and being able to demonstrate they can hold a job.
While i can see the parallels you were trying to draw, that one crucial difference makes a fundamental difference.
Hell, no one in their right might would want to hire you, because they wouldn't want to get hit with the same label and end up in your place as well.
The vast majority of homeless people suffer from mental illness, and have come from broken homes - experienced severe childhood trauma.
These are people who, mostly, never had a chance. This is the key point. They were dealt a truly garbage hand in life, and no amount of 'pull yourself up by your bootstraps' is going to change that.
They need care, they need education, they need homes. They need people to invest years of work into their rehabilitation.
And, in pretty much any developed country, providing that is roughly equivalent in cost to what we currently spend punishing them.
The state is paying the cost of homelessness, but it is doing so in a fashion that both fails to solve the problem and makes the quality of life for these people even worse.
Have you talked to any of them? Most of the ones I've talked to don't want anything to do with work. Yes, work on its own is not enough to rejoin society. But work is a prerequisite for rejoining society. It is necessary but not sufficient. If you don't want to work you cannot rejoin society.
At some point you need to assert some tough love and force them to do things they don't want to do. Like work. Or not drink/do drugs.
Your thinking is essentially the tough on crime mentality: Blindly lock people up until they behave the way you want them to. That's how the US ended up with the largest per capita prison population in this solar system and absolutely fucking nothing to show for it.
Tough on crime has failed. Spectacularly so. I wonder for how many more decades one can keep running in the wrong directions with fingers in their ears, telling themselves that approach will start working any day now.
Worse, your proposed punishment may actually constitute an improvement in living condition to the people you're trying to punish ("Off to free meals and a warm bed I go!"). Are you really willing to spend lots of money imprisoning people to give them what you could have given them for way cheaper, just so you can tell yourself you're "punishing" them? At that point one might as well stop pretending one has anyone's best interests in mind and admit to pettiness.
They probably don’t want to work cuz working low wage jobs is a shit experience and are one bad day from losing their job and becoming houseless again.
"Couldn't find work."
There seems to be a proclivity in the states to solve social issues by diving headfirst into a dystopian nightmare, instead of learning from somewhere else.
That aside, jails are more expensive than simple housing. Just house them regardless. It's way cheaper.
Can you elaborate on the "get a job" part? I don't think most employers would be lining up at the door to hire people who presumably have no recent work history but would either need to be paid enough to be able to afford bay area housing within two months or have to be fired after they're unable to show up to work due to being incarcerated. This just sounds like "throw them all in jail" but with extra steps to try to maintain the moral high ground.