There was a recent study that, like everything else with the homelessness problem, Housing First is most successful in places where housing is not ridiculously expensive to find, so the costs of implementation are low and housing units to place people in actually exist.

The exact same things that burden the private housing sector in the US (excessive land cost, overly restrictive zoning, neighbors suing and constantly throwing roadblocks) also restrict the public housing sector, since the public dollar goes less far, and the public sector has to comply with the exact same laws.