Self-driving technology will overtake average human ability with regard to safety within a decade, but the biggest hurdle will be public acceptance. The AI will not make the same kind of mistakes humans make. So while the aggregate number of accidents will be (likely much) lower without a human at the wheel, the AI will make deadly mistakes that no human would make, and this will terrify the public. A intuitively predictable crash will always be scarier than one that makes no sense to our minds. The only way self-driving tech will ever succeed is if the AI can be limited to the same kinds of mistakes humans make, just fewer, and that's a VERY hard technical nut to crack that I do not believe will be solved anytime soon.
That said, I still believe that the ubiquity of cars is inherently a problem, human operator or no. If we put more effort into self-driving busses and autonomous trains—which have regular schedules, routes, and predictable speeds—I think we would see much greater dividends on our investment and far fewer "unintuitive" errors. Our collective fixation on cars blinds us as a society to this option unfortunately. More cars just clog up the road even more, demand more parking, and otherwise monopolize land use that could be more productive otherwise. More idling/circling driverless cars adds to the blight rather than relieving it. We need to transport more people between points in higher density, not lower, and cars are the lowest density transportation options available.