If it proves to be difficult enough, that level will be when self driving cars are only marginally better than human drivers. That is an unlikely scenario, but it will probably stagnate at a far less perfect level than all but the most pragmatic dreamers envision.
Not before a long time.
A death is really expensive, Americans are trained to sue like crazy, a car accident is a simple case to grasp and attribute to the manufacturer who has a lot of money to pay.
The US has this "anyone can sue anyone at anytime for crazy damages" system. I'd expect any issue to quickly be brought to court and make an example out of them.
Last but not least, after the first death related to a self driving car, the second death will be on the journalists fighting for the coverage.
The first death already happened with a tesla.
Most people who are still in the media saying it can be done have a financial stake in saying those things. This doesn't make it true.
They'll get 90% there and never really figure out the other 10%.