I'm not sure how that is related to gentrification. And last time I checked, people are not homeless out of choice, but from socioeconomic reasons, mental health problems and just bad luck. I'd argue that rent control and housing stability can go a long way to mitigate those problems in the first place.
And you might be partially right, nobody likes homeless people. But for very different reasons. I don't like that society lets people fall, almost permanently, on the side line.
> They are an eyesore and a burden to the community.
The same could be said about SF's tech workers at $social_network_of_the_month. Just because you think you have value, does not make it so. People matter, period.
Abstractly, yes, you are absolutely correct in every detail.
In practice, how much money should that be expected to translate to? From what I've seen, SF could dump literally it's entire yearly budget at offering care, compassion, and generosity to the homeless without the homeless population shrinking.