I saw your comment in my thread about arguing physics in these forums and it made me curious to find the thread you were referencing. So here I am.
Since they only made it through high-school and entry-level college physics, these guys are amatuer physicists that actually believe that tractive forces in real life, away from a textbook, are only relative to the normal force and the coefficient of friction. Unfortunately for them, they didn't go far enough to learn about a term called hysteresis, which is a significant contributer to maximum tractive force, and it does not change the coefficient of friction ("bilinear friction") between 2 given materials. When you introduce an asphalt surface with normal (perpendicular) deformations, and an elastomeric material that allows it to apply that force directly into those deformations, the elementary formula for traction no longer works. However, when you consider this effect, it quickly becomes evident why lowering tire pressure, hence increasing contact patch, improves traction. So, they're wrong to think that this increase in maximum tractive force can be explained away simply due to the "sidewall flex" of the tire. Though sidewall flex can alleviate the shock in the initial launch, and it can delay the application of some of the force from the launch, it cannot actually increase the maximum tractive force that can be applied by the tire. Due to hysteresis, though, increasing the contact patch actually will increase the maximum tractive force that can be applied through the tire.
While they are, for all practical concern given similar tires, correct in their assertion that the contact patch is the same given a normal force and a tire pressure, I'm also 100% certain you are also right that the deformation of the sidewall, and of the surface of the tire will cause a difference in the contact patch area. To make a more appropriate comparison than your pictures, they would be well served to imagine run-flat tires at 16 psi vs. slicks at the same pressure. A run-flat tire is nothing more than a tire with an extremely stiff sidewall, stiff enough to hold the weight of the vehicle without any air pressure. Even at 0psi it will have less contact patch than the slicks would at 16 psi, even though their model apparently predicts otherwise. They just seem to refuse to accept the fact that sidewall stiffness actually does factor into the contact patch area.
Hopefully, after reading the above you will feel a bit vindicated and they will feel a little more open-minded to what you have to say. I'd love it if your PhD friend would care to validate my statements.
Chris