Do you really think people with housing would become homeless in hopes of getting a new place?
If free, safe lodging is available without restrictions, demand will quickly outstrip supply with folks moving from other cities (LA, Seattle, Portland, etc) to fill the available rooms. It's a $50 bus ticket, city officials will likely fund these tickets for them to get to SF to take advantage of this. SF had programs for this exact thing to get homeless out of SF [1]
1 - https://www.vice.com/en/article/bvg7ba/instead-of-helping-ho...
Housing homeless people reduces overall homelessness. If we do it enough, everyone will be housed. Part of this is lowering housing prices, another part of this is housing people who can't house themselves (for various reasons).