I've worked in housing advice and have been personally involved in rehousing homeless people. A member of my immediate family is a director of a housing association and has run several supported housing facilities.
It'd be fantastic if homelessness was as simple as "give people homes", but it isn't. Housing is part of the problem - a very large part in some areas - but chronic rough sleepers have very complex needs.
If you don't believe me, go out and talk to some rough sleepers. Ask them about their life stories. Ask them about the people they've met on the streets. You'll find some people who have had a bad run of luck and just need somewhere to stay, but you'll also find people who are highly dysfunctional and need intensive support to rebuild their lives.
Finland have substantially reduced homelessness, but they're some way from eliminating it. The most effective element of their programme appears to be homelessness prevention, through early intervention for families and individuals at risk of homelessness.
The obvious difference is simply money. Finland spends 50% more on welfare than the US as a proportion of GDP. They have publicly-funded healthcare of exceptionally high quality, extensive social support and a comprehensive public housing programme.
This all connects together. Fewer people fall into homelessness as a result of poverty, because they have a comprehensive welfare state. Fewer people become homeless because of mental illness or drug addiction, because there's comprehensive healthcare coverage and easy access to rehab. Fewer people become homeless when leaving prison, because ex-offenders are referred directly to social workers who can provide personal support, drug treatment and housing. The housing-first model is one element of a holistic strategy to reduce poverty and destitution.
The US could achieve something similar, but it'd cost somewhere in the region of $500bn a year, even accounting for efficiency savings. It also takes a long time. Finland's homelessness initiative more than halved homelessness, but it took the best part of 20 years to achieve. Sadly, I'm not sure that America has the political will to spend a lot of money over a long period of time.