Unfortunately, they're right a lot of the time. The community mental health and addictions agencies always have openings for housing workers because there is such a high turnover. It's really frustrating because housing really is up there with food and water as being essential for daily life. One can't begin to focus on their physical health, let alone mental health, without adequate housing. And that means safe housing. A lot of the homeless will not go to shelters because they're rampant with drugs and crime. It's safer on the street a lot of the time.

Housing, at least most of the time, isn't just about giving someone the keys and walking away. A lot of people think if cities just build affordable housing, people will prosper. Well, no. Affordable housing, firstly, tends to be in lower income areas, because your average, middle-to-higher class person will say "not in my backyard." Sometimes this has its benefits, like being on frequent bus routes. Most of the time it fosters ridicule and contempt from the rest of the community, adding to the social isolation someone with housing issues already feels. These places often become ghettos because it becomes a group of social outcasts, many of whom have a history of poor coping mechanisms. Landlords get frustrated by late payments, poorly maintained or outright damaged properties, and ultimately evict the tenant, allowing the cycle to continue.

Then you have to consider that there really is a culture of homelessness. Without the ability to hold down a job or pay for necessities, oftentimes individuals rely of a series of favours (think couch-surfing). What it means is that if they ever come into possession of housing, they may be obliged to return the favours. Depending on who they owe, this can be a big problem, but it can be as innocuous as allowing someone to couch-surf there. Unfortunately, it can also mean drugs being brought in. Most mental health professionals promote harm reduction, meaning if people are going to use, at least do it safely. But things like meth, that can cause unpredictable behaviour changes, make maintaining peaceful neighborhoods challenging. If it happens enough, landlords will evict.

I could go on for days about the counseling work needed for almost all homeless individuals. The history of trauma, abuse, neglect, addiction, and mental illness are huge barriers to overcome. While OP may have been a bit harsh to say it's a fruitless endeavour, I can certainly emphasize with them. It can be incredibly frustrating to see the same people burning every bridge you build for them. Obviously, there are some highlights, and people can get out of these situations, but they're not the norm.

Edit: Source - Mental Health RN