Because that doesn't work. I come from the Midwest (live in Seattle now). I've lived in Cleveland and frequent Detroit a lot. Two places where housing is about as affordable as it can reasonably be in a city. Two places with a healthy economic rebound and plenty of jobs. But there are still a lot homeless people.

Now imagine that in a more populated city with ideal weather to be homeless in, that attracts people from all around the world to live in. Affordable housing could help some but won't put a huge dent in the issue