That being the case, the value-added of free housing appears marginal compared to the cost. It would best target those who are a) chronically homeless, but b) neither addicted or mentally unfit to be employed, which sounds like a vanishingly small demographic. Mind you I take "effective" and desired outcome here to mean that they won't continue to roam out in the streets, either to be close to dealers or from being mentally unwell.
States with lower housing costs do appear to have lower homelessness rates.
Zoning and regulatory reform spurs more building including smaller, dense units, which lowers prices. There's a fantasy among leftists that developers are snubbing mixed-density and dense mid-rises because condos pay more. No one leaves money on the table. Everything is built on credit, and banks deem these builds riskier in large part owing to the impact of zoning and regs. Condos have a lot of overhead, which checks a lot of boxes, while this would be unaffordable for other units. They also ignore that there are small developers in cities that try to compete, but can't get loans, because of these regulations.