That's exactly it. You have to differentiate between the observable signs of intoxication that you describe, such as slurred speech, unsteady gait, and difficulty walking. People tend to get tolerance to that a lot faster. You cannot get tolerance to some of the complex cognitive processes that are required for driving--divided attention, response to complex driving situations, and certain aspects of vision.
You just can't get tolerance to that, even if you are an experienced alcoholic.