Sure they can. During snow squalls, accident rates obviously go up (especially if the roads are icy or if it's the first one of the season), but most drivers are good at assesing risk and take appropriate actions.
Importantly, they have the ability to decide that it's worth the risk to go out on the roads to pick up Sally from daycare (for example). In any case, if the car computer decides it won't drive, do you really think people aren't going to drive the car manually so that Sally can come home for the night? Of course they will. And they'll be fine.
Have you ever been on the freeway during white-out conditions before the ploughs have gotten a chance to come by? People slow down, turn on their hazards if necessary, and if the road markings aren't visible, naturally form two lanes instead of three (straddling the lanes). In snowy climates, a simple snow squall is something to talk about, but not stop you.
People are awful in these conditions and exercise incredibly poor risk decisions.
Sure, a bit of rain doesn't help the safety stats, but you can't just put the whole city on hold when there's a downpour or some fog.
I lived in snowy climates for a decade. Your summary completely disagrees with what I experienced. In whiteout conditions there were always cars that had gone off the road. Even after years of driving in the snow, people get into accidents far more often than in dry conditions.
And don’t forget: most of the US population has little to no experience driving in winter conditions. If the Bay Area got an inch of snow, there would be more collisions than a demolition derby.
Businesses don't shut down just because the forecast says it might snow. Plows can't get out fast enough to clear the road to a point the lines would be readable/visible for a self-driving car. Humans may not be perfect but we find a way to get home in the snow without the massive casualties you're implying should be happening.
It seems as long as these machines are safer on average than humans, they're worth deploying.