That is where you are dead wrong. I would highly recommend you walk down 13th street (aka Duboce or Division, can't remember) and talk to the people living in the tent colony lining it. I have; it used to be my way to work till the hassle became too much and I started taking a detour. I would posit and say 100% of them are there by some sort of choice (not following the rules of the shelter is a huge #1). Drugs, alcohol and theft (bike, specially) are rife there.
I live in SF, and just can't stand the holier-than-thou approach taken by those who have not experienced SF's homeless firsthand. These are not your "lost my job and house and am forced to live out of my car" homeless; these are people, most of whom made a conscious decision to come to SF because they're attracted by the generous benefits, or got hounded out of whatever city they were in.