No, it wouldn't necessarily have to. Self-driving cars are going to fail in ways that seem very stupid to us, but they might handle "normal" accident-prone situations much better than humans and thus be much safer overall.
You can see that at the point where the Tesla busts through the stop sign - any normal person would associate a stop sign with danger and proceed with caution, except of course the teenager with L plates.
Current AI is better than a drunk teenager, but that's too low a bar. Driving sober, well-rested and in daytime (driving during the day also keeps you away from all the other drunkards) is actually remarkable safe. Remember that the safest mode of travel per journey is not the airplane, it's the motorcoach.