>trusting your kid's lives to a computer, the same technology that crashes on the desktop, and on our smartphones

This is all math here. If a computer is statistically better at driving than a human you can ask the same question as "Would you give up control to decrease the chance of an accident? Or would you risk your kid's life just to feel in control?"

>What happens when the first car has a BOSD and the backup human driver isn't paying any attention and the accident happens within seconds of the crash?

People will probably die. The the same thing that happens when a human crash a car which is already happening everyday.