I was under the impression that 99.9% of people take their cars out on publicly owned roads when they drive. Am I mistaken?

But this would go in the cars, not in the roads, so that's irrelevant.

And the TSA rules apply to my suitcase, not to the plane. What's the difference?

Your suitcase isn't required to have a built-in X-ray machine.

Well it wouldn't preclude your use of the car (you can still enter it and turn it on), you just can't drive it on public roads (Essentially the entire use-case of moving a vehicle).

TSA is at the airport lobby and technically only precludes me from entering the airside of an airport, but its purpose is to prevent me from getting on a plane with a gun.

I don't think point of implementation is a strong argument here.

Wouldn’t an closer equivalent be DUI checkpoints staffed by government-accountable law enforcement officers at the entrance to the road? Not an automaker forced to privately police you in your own vehicle