Homeless people live where they do for the same reason renters or homeowners do: friends, family, history, allure, and so on. If, say, Jackson Hole, Wyoming, announced and financed the biggest homeless benefit program in history (they won't, but bear with me), then would California's homeless people leave their warm, blue skies? Would homeless people in New York uproot and move halfway across the continent?
As an aside, my idea wasn't to take homeless people in SF and transplant them to LA, but to edge them out to elsewhere in the Bay Area. If the whole region took on SF's lead, then the other counties would do the same until there was a location with either enough space/tolerance for the homeless, or a population that couldn't or wouldn't join to pay in the same benefit scheme. And if they want to move back, they can use their regular income to help them get a job and a place to live.
Interestingly, the SF Gate reported that there are around 7,000 homeless people in San Francisco. If each of them were paid $500 each month to avoid the city, that cost would tally to $42 million per year. For the legions of wealthy tech workers and venture capitalists in San Francisco, that would barely be a drop in the bucket. For cleaner, safer, and more pleasant streets, is that not a worthwhile investment?