How does sidewall stiffness matter? The ground is holding the car up; the air is not.

I can drive my car safely at 55mph with 0psi in the tires. I believe this methodology would suggest my car is weightless in such a situation.

The quickest way to realize the contribution: Imagine the tire is made of steel so it does not deform when the air is let out. This would be terrible to drive on, but it would not require inflation even though it's hollow like a rubber tire. A rubber tire will usually not "go flat" even with the rims/wheel installed even without pressure (aside from atmospheric which is present when it's put on the rim). So some load is supported by the structure of the tire itself through the side walls. How much will vary with the tire.

In the case of a perfectly rigid sidewall, the internal air pressure could be 0 and the car would still be held up.