The best measure to reduce homelessness is to provide timely support for people who are being evicted from their homes before they lose their jobs (which they might still have) and before their mental health deteriorates. This is the point at which assistance is most effective. You have heard the saying, "an ounce of prevention is worth a pound of cure". Such programs have been applied to great effect in e.g. London.
The way to respond to people who have experienced chronic homelessness with complications is different and more difficult.
Unfortunately this is pretty selective evidence, but I know for a fact they don't exclude people on the basis of having mental illness or addiction problems, I've worked with them personally.
The problem is also political, unfortunately. $10 billion isn't going to change zoning laws and a NIMBY attitude of freezing things in time.
If they like that point in time so much, they should build a museum, sheesh!
But that doesn't answer the question. It's "a lot" of homes, but less than a tenth of what would be needed.
And you need a bunch of social workers too at minimum.
> And you need a bunch of social workers too at minimum
Ok, sure. Remember, we're spending the $50B that's been lit on fire so this gives us more jobs and a happier country. And that money circulates in the economy rather than expatriated profits by the defense contractors.
This comes with the giant caveat that we exclude the external costs of such a huge project, like social welfare visits, probation or monitoring if needed, or even just placement programs. Likely those all combined would be a third of the total cost.