The issue is prison is about "doing time".
If someone doesn't want to "get better" then that is the thing you treat.
I didn't watch the video mostly because I don't know what it contains and I have kids around.
Making the assumption that it's about people who want to be homeless, honestly that's fine. There's nothing inherently wrong about being homeless if that's what you want. The problem is people causing public health issues because they defecate in public areas, tear apart trash bins, or cause other issues like attacking other people... none of which is acceptable to society.
These people may not want to be part of society, but they are living where society exists.
If you have a chance to watch the video, I'd recommend it (it's about 1h, as a heads up). It is certainly mature themed, so may not be suitable for younger viewers, but it is a news station-produced segment/doc (KOMO in Seattle), so is not wildly inappropriate, and I found it to be a fascinating and hard look at a real problem. (In fact, the first thing I did when I saw this thread was ctrl+F "yout" to find out if anyone had already posted it).
The video explores the visibly-homeless issue in Seattle, as well as a program to address the problem that was implemented in Rhode Island (specifically, enforcement + long-term treatment). I found it worth the watch.
What do you do in cases like that if prison is off the table?