SF is in Northern California, so I don't understand the question.

If you mean to ask why the homeless don't go to Nowheresville, CA, it's probably because there are no social services available for them there.

I'd say SF is more mid-California, compared to what's north of Sacramento. North of Sacramento is pretty rural, full of people wanting to implement the State of Jefferson. But you're right, there isn't much in the way of social services in northern California, due to the population. The only real population centers are Chico and Redding. I haven't spent any time in Chico, but Redding has quite a few homeless, and there isn't much in the way of social services. The City of Redding doesn't have money to burn like SF apparently does, so the homeless pretty much have to fend for themselves. I get the feeling that a lot of Redding's homeless population is on their way somewhere; Redding is kind of a midway stop along I-5 or to the coast.

I'm on the East Coast, but Wikipedia describes San Francisco as "the cultural, commercial, and financial center of Northern California."

Yeah, and it's understandable why they would say that. Realistically the only places people care about in California are Los Angeles and San Francisco (and to some extent Sacramento and San Diego), and compared to LA, SF is definitely north. Culturally and politically, though, north of Sacramento is quite different from SF. I mean, you see lots of signs supporting the formation of Jefferson State (and seceding from CA). I've had California natives talk about how SF wasn't really northern CA. And geographically, it actually is kind of mid (on the northern end of mid, but still).

Northern California winters aren't harsh, but still don't compare with Southern California's.