This is a major misconception. The people you see homeless on the streets are very disproportionately those with mental illness(es) and addiction problems. But studies show a huge percent of the homeless have jobs; the link below says 40% to 50%.

But we don't see people workings and living in their car or in a shelter "being homeless," so we tend to think of the visibly homeless as representative of all homeless folks.

And for those who do have mental illness or addiction problems, well, those problems are severely exacerbated by being homeless. They'd be more likely to get treatment and improve with housing.

https://www.usich.gov/guidance-reports-data/data-trends