So, I've never been homeless, but I do work in mental health. We see a lot of patients that are chronically homeless, and they view shelters as a place to dip into briefly when they have to show that they've got a physical shelter (so that we can do a "safe discharge"), and then promptly bounce. Some won't even make it to the shelter, and have their ride from the hospital pull over and let them out so that they can head out immediately.

There are caveats to be added to the above, but, I've never gotten the impression that our implementation of shelters has resulted in something people utilize in anything less than a crisis situation, for a population that doesn't consider "chronically homeless in the winter in a northern latitude" a crisis.