That's a well-documented concern in aviation. Hard to imagine it will be less prevalent in a population that doesn't even need to get re-certified now and then. General skills deterioration is probably not the issue that having to take over very quickly is, but it is one.

I wonder how much more the average driver's skill can degrade. We never re-train so isn't there a natural degradation already? Will automation make that worse or will it not be significant?

For the typical (certainly US) driver, the initial training is just to get to a minimally viable set of skills so they can pass their driving test. The vast majority of people aren't taking performance driving courses to get their drivers licenses. I'd pretty much guarantee that almost every driver is more skilled 10 years after they get their license.

I cannot comment on driving experience/skill since I don't have a driver's license but I frequently observe drivers without a working understanding of the traffic rules and signs, even though they once learned that in the theory classes.

I'm not sure that experience implies skill but you make a good point.