In the US, the political will for "eliminate homelessness by giving them homes" is a political non-starter in a lot of places. In the left-liberal places, you've got entrenched interests fighting tooth and nail to prevent any housing at all from being built, and in right wing areas you've got a "being poor is God's punishment" attitude pushing back against any positive change at all. To make matters worse, any effort to solve the issue results in a large influx of homeless people, because when the standard across the country is one step above "hunt them for sport", you come across as a haven.
In the places it has been tried, housing-first solutions have been wildly successful. It turns out getting your shit together is a lot easier when you don't have the constant stressors of living on the street.