I'm sympathetic to the argument you're making, and I've even used it a couple of times when talking about stuff like browser standards, Facebook Basic Internet, etc... It's not a dumb argument.
But at the same time, homeless people don't care about your theoretical utopia. They need help right now, and there's a very real cost to saying, "nobody should help this person suffering until we can do it right."
There's a balancing act here. Unless private care is going to make it harder to solve homelessness in the future, we should embrace it as a band-aide solution while we work on public infrastructure. Because let's be honest, public solutions like guaranteed housing are likely pretty far in the future -- I don't see a lot of evidence that there is enough political momentum right now to make that happen. The evidence I see suggests Medicaid has a decent chance of even getting worse in the immediate future.
We need band-aide solutions right now.