That's just not true. The problem is that government housing is so terrible that sleeping on the street is preferable to some. Public housing in Austria is high quality, cheap and plentiful, with two-thirds of Viennese residents living in it. Coincidentally, homelessness is a non-issue in Austria compared to the US (based on my experience living in the Bay for 10 and in Austria for 20 years).

See this video for what government provided housing could be. https://www.youtube.com/watch?v=d6DBKoWbtjE