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.
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.