I think you need to account for the fact that SF has a population of ~860,000 where as NY has a population of ~8,500,000. You also have to account for the weather in SF vs NY. Not defending either city just trying to put the numbers in to perspective

I was accounting for that. NYC manages to provide shelter space for 10x the number of homeless persons as there are total in SF, while having 10x the population of SF.

I'd argue that it's easier for a smaller affluent city of 860k to solve a problem of roughly 3,000 +/- homeless people (their problem isn't the 6k overall figure, it's a smaller subset of that), than it is to deal with 10x that homeless figure at 10x the city size. Human problems like that get more difficult with scale rather than easier.

I'm not sure how SF weather plays in to providing shelter space for the homeless as a problem. It should be easier and cheaper for SF to do it, as SF requires both less air conditioning in the warm months and less heating in the cold months. Of course, this is also where SF's terrible zoning laws show up as usual.

The weather effect isn't about the cost of the shelter space. There is less incentive to put up with shelter rules (or risks, as the case may be) when the homeless can sleep outdoors somewhat comfortably year-round.

Anecdotally I can tell you shelters begin to fill up when it snows or gets extremely cold. This weekend NYC is looking at lows in the teens. SF, lows in the mid 40s.