A fully autonomous self-driving car (if that is what you mean by self-driving) will have to be capable of reasoning it's way out of every possible situation, which requires general AI.
The sheer number of cars used and the variety of places they're used in means that thousands of cars get caught up in bizarre situations every day. Cars will be shutting down inconveniently all over the place unless they can apply general reasoning skill to navigate through the innumerable trials that drivers currently have to cope with.
I can think of a ton of everyday problems that will require actual AI: Following detour signs. Understanding a cop's hand signals. Understanding crossing guards in school zones. Making room for emergency vehicles on crowded streets. Driving on dirt and gravel. Navigating parking garages (ever notice that signage in garages isn't standardized? Probably not, because you're not a robot).
Most of the self-driving crash videos I've seen involve a car that lost it when it couldn't follow the white line. Following the white is not good enough.
The sheer number of cars used and the variety of places they're used in means that thousands of cars get caught up in bizarre situations every day. Cars will be shutting down inconveniently all over the place unless they can apply general reasoning skill to navigate through the innumerable trials that drivers currently have to cope with.