Anecdotally whenever I hear a SF resident say the "crime problem is overblown", they almost always live in the nicer areas where there's virtually no crime. I lived on the border of the Tenderloin years ago, and saw things that I've never seen anywhere else. I knew a guy who was murdered in the Polk Gulch, I had my car broken into (and rummaged through multiple times after I started leaving the door unlocked), I had to physically assert myself with my friend to not seem like a target when a man brandishing a metal rod was eyeing us suspiciously, I stepped on a used hyperdermic needle which pierced my shoe but luckily did not prick me ... this is only what comes to mind at the moment as a write this, but I'm sure there's more incidents I experienced that I could conjure after some reflection. Is my experience anecdotal? Sure, but I lived it. Your experience with the city has likely been a much better one, free from such disturbing experiences. Does that mean I didn't experience this though? Or that it is "overblown"?
tldr; I lived in SF but now happily live in New York.