I am not familiar with the statistics, but I used to work with people who are homeless regularly, in-person, so I can answer a couple of these.
Do you mean that those suffering from addiction are incapable of modifying their behavior in response to external conditions?
Well, think about it this way: for people who become homeless as a result of addiction, getting kicked out of whatever housing they were in before becoming homeless obviously wasn't enough motivation. For people who become addicts after becoming homeless (which as someone mentioned up-thread, frequently happens)... well, they're already homeless. Also, lot of the people in the latter category, in my experience, are barely more than children.
On a more philosophical note: Hardship is a motivator, until you reach a threshold. The threshold is when you become overwhelmed and discouraged; it is no longer apparent that your actions can influence the outcome. People feel trapped. Despair is a state of being; even animals experience this. People on the street are nearly always sleep-deprived-- lights, the threat of assault, cold, wet weather, being moved constantly by police. They're also chronically afraid, eating extremely tilted diets, and are just generally worn down in a way that most people can't even imagine. Shelter offers them the opportunity to get their brains online in a way that it's hard to manage when you are sleeping rough. To be energized, effective, make good decisions, etc. from that place is a massive undertaking.
How successful is housing at reducing the other costs you mention?
Ask an EMT how many times they respond to emergencies relating to homeless people. Exposure injuries and deaths are very real and costly. Plus, just the cost of having police go out and move homeless people away every time someone calls, respond to calls about the belongings of homeless people, etc.
One of the biggest fears guests had in our shelter, which was very low-barrier-- almost anyone could come in and sleep on a kinder mat, even if they were drunk or high, as long as they weren't disruptive-- was other homeless people. Getting stabbed, sexually assaulted, robbed. All these are things that public services address that, as you will notice, happen a lot less if you have walls around you and a door you can lock.
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If overcoming addiction is the linchpin of the homeless person's other problems (as I presume it is), and housing doesn't make the addiction go away (as sadiesortsitout says), how does the housing play into that person's rehabilitation(?) process?
I don't know about the linchpin. Housing doesn't make addiction go away, but it provides a stable base from which rehabilitation begins to be possible. Homeless people are sleeping wherever they can, with whoever they can. If you're trying to get off drugs, but sharing a tent with someone who is using drugs, you're not going to get off drugs. If you're freezing and someone has drugs that will take your mind off how cold you are, you'll use drugs.