Most shelters do in fact allow people to bring their belongings with them (within reason). Some even provide storage space, and the city provides a free self-storage facility.
Prop F addresses CAAP (cash welfare), not housing. You don't have to be receiving CAAP to qualify for housing assistance, and you don't have to be homeless to qualify for CAAP.
SF has been struggling with a chronic homelessness problem for decades. Offering voluntary services does not work. To put in in Trek terms, the problem isn't the gimmes, it's the ghosts and dims. Gimmes are easy to help because they can act on their own behalf and in their own best interests. They accept services and don't end up being chronically homeless. The ghosts and the dims, on the other hand, are a different story.
Is sweeping their encampments an ideal solution? No, far from it. But what else is there for us to do? Let them languish on the streets? Honestly, what would you have us do?
Housing First has been the policy in San Francisco since 2008, and state-wide since 2016.