I think he nixes your case at the start, when he goes through the sanity check thinking about the flat tire. Also, I think your rigid body case use the walls of the tires to apply force to the care, not the force from the pressure to keep the car static, which is why it fails.
I agree that higher tire pressure results in smaller surface area, but it's not clear to me that the tire itself does not support the car, skewing the result. So I'm not convinced this sanity check is sufficient. For example, as another commenter mentioned, run-flat tires can support the car on stiffness alone.
I agree, the sanity check is not sufficient. It confirms the concept that as pressure decreases, area increases, but is not sufficient to confirm a linear relationship. In the limit, the area clearly does not become infinite when pressure drops to 0.
For the case you suggested above, this is essentially how run-flat tires work. Of course, the sidewall of the tire isn't as stiff as the support ring, but they do provide some support.
He mentions the flat tire, but he ignores the fact the pressure of a flat tire is 0, leading his measuring method concludes that the weight of the car = F = PA = 0A = 0. His measuring method flunks his sanity check, but he never notices.
Let P_m be the pressure recommended by the manufacturer. A better sanity test would be to fill the tires to .75P_m, then to 1.5P_m, and see if the area in contact with the ground has shrunk in half.
With a completely flat tire, the rubber sandwiched under the rim will transfer weight directly from the rim to the floor, so the pressure equation won't apply.
