Do you have evidence that most people aren't above-average drivers? I would be pretty shocked if driving skill was normally distributed. I would guess that there are few extremely good drivers (you can only do so much to improve), but significantly more extremely bad drivers. This would drag down the average.
Even if you word your survey question in terms of medians, people are bad at numbers. I don't think you can really decide if that's illusory superiority without more digging.
I don't think there are huge numbers of extremely bad drivers. People who are extremely bad drivers tend to have their license taken away, and thus are no longer drivers at all. In fact, I would wager that, when considering average as the mean (not the median), most people are probably below-average. As previously mentioned, there's a minimum bar for driving ability, and I would also wager that most people are pretty close to the average to begin with. But there are fair number of exceptional drivers (e.g. anyone who drives for a living is likely to be a much better driver than the average person, as are any of the numerous people who do a lot of recreational driving).
All that said, even if your premise was correct, it would allow for a simple majority to be better than average, but it wouldn't allow for 80% of people to be better than average.
> extremely bad drivers tend to have their license taken away, and thus are no longer drivers at all
Your optimism about rule-breakers' adherence to the rules is heartening. A lot of that long tail of bad drivers we are talking about have no issues driving without a licence...
Not to mention that "better driver than average" is a very fuzzy definition, and more than 50% probably hit at least one of the possible categories, e.g. safer? Quicker reaction time? More pleasant to other drivers? Most comfortable to ride with? More technically skilled when it comes to difficult conditions? Able and willing to go fastest? Gets the fewest speeding tickets? Etc.
Even if you word your survey question in terms of medians, people are bad at numbers. I don't think you can really decide if that's illusory superiority without more digging.