Let's start here. What are the steps needed to get the state to do this? Is the money for it taken from other places, or by increasing the deficit, or by raising taxes? How do you convince other politicians of your plan? Where is the land taken from?
Again, this seems only like a first step in mapping the problem.
If you agree to the end goal, you can pick whatever answer you like to your financing question, it's not really more important than “what color should the buildings be” as the cost would be a small fraction of say IRA or the yearly defense budget, or Trump's tax cuts.
The legitimate question being: what to do with the plutocracy, and to that question I'm afraid I don't think you Americans will see democracy ever again untill you are willing to get back to your revolutionary roots and take it (or at least make the oligarchs scared enough to give up part of the power, which is the best case scenario TBH).
There seems to be a fundamental misunderstanding in this discussion, because to me it seems like we're jumping from "this is really simple" to "it can't be done" and back. Which is it -- is it simple, or is it so complex that it's impossible?
1. Many problems, including homelessness, are relatively straightforward to solve for a willing State entity in a developed country. Everyone in power that says otherwise is simply lying. This is the “this is really simple” part.
2. The people that get elected and are ruling are not willing to fix these problems. And the reason they aren't is that it's really hard for someone who is willing to actually get elected because of the plutocratic nature of the system. This is the “really hard” part. It's not that it's complex, it's that the interest of the people is fighting against the ideology and economic interests of the ones who have the political power to get someone elected.
See, there's no contradiction at all, and reminding people that “yes, we can in fact have nice things” is actually a political strategy to try overcome #2.
Build housing would be a great start! Markets don't care about low-profit cheap small apartments, so there's no supply and the demand isn't being met, so it's the mission of the state to step up and compensate this market failure.
Then you can also tax properties based on their occupied land area, which incentives the private sector to build higher density, which is much cheaper in terms of infrastructure cost and cost of living as well.
Then you'll stay with homeless junkies and vets, for which you'll need social workers and dedicated structures, but you could also try fixing your social issues and wage less wars and the number of people to take care of will plummet and become more similar to what you can find in other developed countries.