Fascinating discussion here and one that is hard to wrap ones head around as it seems counterintuitive. I remember the same experiment in a college physics class and it always amazed me that surface area had no affect on the outcome. I think someone might have hit the nail squarely on the head when they suggested that a skinny tire and a fat tire with the same loading would work equally well initially but the skinny tire would soon experience melt down changing it's chemical properties along with it's coefficient of friction.
It would be interesting (and entertaining) to see a top fuel dragster equipped with both types of tires and data log the first few miliseconds of acceleration. It might turn out that skinny delivers the exact same acceleration characteristics for a split second then goes into early melt down mode and starts spinning wildly leading to further destruction. The fatter tire will absorb the huge power loading and live, hopefully to the end of the quarter mile. In the end, the fat tire has an advantage at the end of the track but not at the beginning. Only a sensitive gmeter could record such an event as it would occur over the briefest period of time.
So as I see it, the physics don't lie, there are just mitigating factors in the real world that come into play.