Where did this institution even come from? Surely there were cars and drivers before DMV's?
If we retained the need to have liability insurance, which protects property rights, I see no reason why insurance companies would not ask for you to prove some competent level of driving. It would be in their interest to know the level of competency they were insuring, and they would probably give breaks to those who do well on written and driving evaluations. Insurance carriers would likely either perform those evaluations themselves or independent agencies they approved of would emerge to specialize in this. These insurance agencies would surely tie a network together with police stations to verify coverage claims, and this still would allow for the court system to suspend driving rights when applicable.
Anyway, off to the DMV.