I have to have fun with this one and disagree. A flat tire places the rim in contact with the inside of the tire. In this case, my theory is void because now you have the rim directly contacting the ground, with a thin piece of rubber between the two. The car is no longer supported by the air in the tire.
You are right. The force the tire exerts on the ground IS a function of the vehicle's mass and that mass's interaction with gravity. BUT... gravity is constant on this planet, and so mass times a gravity constant is called weight. Unless we are taking our vehicles to Mars, that " function of the vehicle's mass and that mass's interaction with gravity" is more simply referred to on this planet as 'weight'.
Unless the ground is propelling the vehicle upward, the 'weight' of the vehicle on the ground must equal the force the ground exerts on the vehicle through the tires (remember that "for each action there is an equal and opposite reaction " issue?). If I weigh 160lbs and my shoe has a 20 sq in surface area, I exert 8 psi of force on the ground, and the ground exerts 8psi of force on my shoe. If your car was able to balance on one tire only, and the surface area of that tire on the ground is 30 sq in, and the psi in the tire is 60psi, assuming that little weight of the car is held by the sidewall in compression(which is a reasonable assumption - you can pretty easily flex a sidewall between two hands) then the only true supporting structure for the weight of the vehicle is a column of air between the rim and tire. And ROUGHLY then, the weight of the car is 1800lbs.
Fun discussion

Fun banter

Reminds me of old physics classes!