I'm somewhat aware of the scope of the issues. Mental health and addiction are often overwhelming when you don't have someone to help you with them, and so a case worker can definitely help. But one of the big reasons common solutions don't work is they're not comprehensive solutions.
For example, if someone is mentally ill, maybe they just need medication. But still don't have any income and perhaps no housing. What happens if they get mugged, or lose their meds, or forget to take them? Now they're unstable again. They need housing, and some small bit of income, food, security, and assistance. These are all resources, and providing them _all_ gives someone a much better shot at getting better and staying that way.
But it's also a misnomer that most homeless are addicts or crazy. One out of every 30 children in America were homeless in 2013, a disproportionate number being LGBTQ. 12 percent of homeless are veterans. In 2009 there were >535K homeless families. In fact, mental illness and addiction are the third and fourth most common reasons for homelessness.
Homelessness disproportionately affects children, destroying their education and keeping them in a cycle of poverty. If we focused more on providing them an education as well as food and shelter, this would have a big impact on outcomes of future homeless. Is providing these things simple? No, but the decision by the population to commit to actually providing them, is simple. The work will get done if we decide to.
We went to the moon on a whim. Surely we can feed and educate some kids.