How much does tire sidewall stiffness contribute to the force holding the car up? I know that can vary a lot. Some cars have run-flats that won't sag when they lose pressure (or won't sag as much, anyway). Low profile tires probably transfer more of the load through the sidewall than doughnut tires. And of course the contribution from the sidewalls (as a percent) will change as tire pressure increases.
Oops that's a great point. I completely forgot about sidewall stiffness. My messing around with treads was probably just me adjusting things to make myself look good then!
On the other hand, if you had hollow steel "tires", the stiffness would be everything and the contact patch would change only imperceptibly with a change in tire pressure.
Car tires are obviously somewhere in this range.