While it's true that you're not guaranteed a shelter bed in most cities, in practice shelters run far below capacity. You can get detailed data on population and shelter capacity here: https://endhomelessness.org/homelessness-in-america/homeless...

For example, it appears California is maintaining around 20% capacity utilization in shelters statewide on average.

Talk to the workers, and they'll tell you the main struggle is not running out of space, but getting homeless people in the door, because as you point out, shelter life sucks. You're absolutely right, one doesn't "live" in a shelter in any meaningful sense. They're not a permanent solution, and hardly even a temporary one.