The homeless can be classified into two groups: temporary and chronic. In any given year, temporary is about 80% of the homeless. We have a okay track record at helping the temporarily homeless, and lower housing costs would make it both less prevalent and cheaper to address.
However, the chronically homeless, at ~20% of the population, take up over 50% of the budget. They aren't able to live independently, regardless of any reasonable cost of living. They're also the most visible group.
We should keep in mind that the solutions for the two are vastly different.