It's not like they magically respawn or something. There are people who have studied the root cause of homelessness, and: it's housing. Sure, other factors make things worse, but there's more homelessness where housing is expensive.
Which stands to reason: when a thing is expensive, fewer people can afford it.
That's why, ultimately, the biggest YIMBY win is going to be not helping out people currently homeless, but stopping the pipeline in the first place by having enough housing.
There's so much evidence from research on unhoused populations and housing-first programs in the US and around the world, but you've got "why wouldn't they" on your side, so all of that evidence should probably be ignored.
No, but people are mobile, and the houseless are also people who communicate through the internet and share where assistance is available.
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]
Which means every city should do this to reduce their homeless population. The "don't do it because homeless people will move from elsewhere to get this benefit" is a race to the bottom.
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).
You are making something of an induced demand argument that basic services for people who can't care for themselves will generate more such people. Fortunately, most people of sound mind understand that living in a former Motel 6 is not a good life and in practice the number of people who are "incentivized" by it are much less than the number of people taken off the streets by shelter. That's why cities with robust shelter programs like Houston have far fewer unsheltered homeless people.
What happens next? Just no more homeless?
No, more homeless will appear. They will fill the same corners and tents that you just cleared the month before.
This is not a solution. This is a grift.