For my book just out (AI Fairness and Beyond, but it's priced for practising lawyers and libraries so I doubt it will be on anyone here's Christmas list) I tried to work out how close the quality of driving is, using data from California which has seen the most extensive use of self driving cars on the roads.
My analysis suggests that in 2019 the quality of driving was roughly equal - self-driving cars caused slightly more reported accidents than human drivers, but with less damage and injury (and it was probable that many human accidents were unreported whereas all self driving car accidents had to be reported to the regulator). Deaths per 100,000 miles were certainly lower than for human drivers. [Sources for the analysis: Jun Wang, Li Zhang, Yanjun Huang, Jian Zhao and Francesco Bella, ‘Safety of autonomous vehicles’ (2020) Journal of Advanced Transportation 1; US government statistics, crashstats.nhtsa.dot.gov/Api/Public/Publication/813060.]
I'm confident that self driving cars have improved since then, and that human drivers have not.
Societal acceptance lags behind statistics, of course, but it gets there if the technology proves itself safe enough for humans to feel comfortable with it. Remember that trains were seen as far more dangerous than horses in the early 1800s (at 40 mph humans would obviously be unable to breathe), and bicycles were too dangerous for women to ride because they would become infertile. More recently, the first fly-by-wire passenger aircraft caused a similar outcry, and even ABS on cars was seen as suspect.
I'm confident we will see self-driving cars on some UK roads within around 5 years. From what I know, rural roads are a special challenge and might be quite a lot further away - surprisingly, the technologies being developed to allow drones to fly in uncontrolled airspace might be a possible route towards self-driving cars on rural roads.