I think Ford is the first company to really be trying to test in snow at all, but that's mostly focused on mitigating the problem of snow on the ground. The sort of snowstorms I've driven in, I don't expect a self-driving car to be able to do in the next sixty or seventy years at least.
under normal conditions, self-driving car systems are tracking white\yellow lines on the road. I am not sure what they are going to do when it is all white during snow. I know GPS is also used but it might not have latest info about road directions.
Furthermore, you don't just blindly follow marked lanes (even if you knew exactly where they are) in a snowstorm. You may follow tracks left by other vehicles, deal with the fact that two lanes in cities often effectively constrict to one, etc.
No, lots of people don't do a great job whether it's because they're driving too fast, not leaving enough room, or slamming on the brakes. But there's a huge amount of judgment and situational awareness needed in bad weather--especially snow.
Certainly, autonomous systems can (and almost certainly will) be restricted to certain roads/types of roads in certain weather conditions and still be useful driving aids. But as soon as you restrict systems to only working some of the time, you've effectively closed off any use that doesn't allow for having a competent driver behind the wheel at all times.
There's a miltary anomaly recognition system that has a person with sensors attached to his head, and he gets shown satellite photos in quick succession. His conscious brain might not notice things, but his subconscious sees things like man-made structures, and the sensors see this, and the images that cause stimulation are flagged. I wonder if a self-driving system would be made that uses the brain's image recognition facilities.
I would guess the person scanning those images would need to concentrate instead of talking to someone on the phone ot next to them. So the next step would be to outsource the required concentration to brains in Elbonia...
I don't think anyone's trying to do it presently, but if the car was creating a virtual 3D map of the road surface as it was driving, it could conceivably learn to drive in the tracks of the vehicles in front of it.
But yeah, a car without a steering wheel right now wouldn't do a lot of people much good, if the car only works in pleasant conditions in daytime.