Given the current electorate in the United States, I think it would be nigh-impossible that human-driving cars would be banned. Notwithstanding the fact that the cost to equip every vehicle on the road with self-driving technology would be wildly expensive in the short term, Americans respond vigorously to threats to freedoms that are ingrained in their culture (visceral freedoms, not freedoms that are less tangible, such as electronic privacy). Examples: firearms, tobacco, alcohol.

I could foresee this happening in one or two Western European countries, though I would expect Germany and Central Europe to hold out (the German electorate resists regulation that impedes the speed and power of their automobiles).

An aside: how would a car being self-driving affect the owner's ability to direct it wherever he or she pleases?