I actually think there are tons and tons of "homeless" people who are completely invisible because they're otherwise normal people who just happen to be living in their car or a tent for an extended period. The people I've met who have been in that situation were not dealing with addiction or psychological problems. They were simply victims of circumstance. In at least one case I'm aware of the person was kicked out by her husband with no job and no address and was totally unable to access any social services. Her getting housing enabled her to get a job and access services she didn't have access to when she was homeless.

I think giving people housing is a huge part of getting them off the streets, and I think you underestimate the number of people who are homeless because you're not seeing the countless homeless people who are keeping their lives together in spite of not having a permanent address.