Where I see autonomous driving going is that they get so good at solving so many well-defined problems that they eventually match and then surpass the safety record of humans, but without ever having the sort of insight that humans have. They will be able to prevent many of the kinds of accidents that humans have, while being subject to kinds of accidents that humans seldom have. What we care about are the overall number of injuries and deaths. But without AGI (which I don't think we'll have this century) progress will be incremental. I don't think we're going to get a re-write that suddenly "solves" autonomous driving. I think we'll get gradually fewer and fewer disengagements until we reach a level of safety that we agree you're safer in an autonomous car than one driven by a human. My best guess for that is ten years, give or take a few.
Agree with that except your best guess of 10 years. And not because I think the cars are going to get better fast (it will be asymptotic) .. but because, on average, humans are such crummy drivers and present a very low bar for the car to better. Sure, we can all think of "stump the car" scenarios, but at the end of the day most driver caused accidents are the result of carelessness and/or inattention, something the car doesn't suffer from. So yes, we will get rid of most of the minor accidents, and many of the major ones. But just occasionally the car will do something brain-dead and kill someone. But even seat belts and air bags so that once in a while.