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!
It's useful to consider the extremes. If your car was somehow supported only by party balloons on the rims, then their stiffness would contribute nothing and the calculation would be spot on.
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.
