Interesting e-420. But that formula only works for balloons that have no inherent sidewall rigidity or a fixed tread width. With a balloon contact patch increases in proportion to the load placed on the balloon and in addition the contact patch increases in both width and length as pressure is reduced or load increases.
Tires have a rigid tread width and strong sidewalls that hold the patch width virtually constant throughout all loads. The sidewalls will buldge under pressure and the patch will grow in length only. Also, the contact patch area increases much less in proportion to the load because the tire structure itself will force tire pressure to increase with the increasing load resulting in a reletively small vertical deflection of the radius of the tire sidewall. Thus, large increases in load result in significantly less increases in contact patch area than a balloon would exhibit.
I think you hit the nail on the head though on contact patch shape regarding hydroplaning and pushing through snow. Narrow tires expell water better and cut through the snow more cleanly.
I agree with you completely that contact patch width will remain virtually constant with changing loads and will change in length with load and pressure.
I also agree that for a given tire, large changes in load or tire pressure will give a somewhat non linear contact patch area as you go through extremes in sidewall deflection and tire pressure.
When comparing two tires of different aspect ratios on the same vehicle with the same tire pressure, the effects you mention have less of an impact.
I believe, and have verified (see comments below) for a given vehicle, and the SAME tire pressure, a 75 series tire and a 50 series tire will have very similar contact patch AREAS for the tire pressure common to cars, 30-40 psi +. (Wildly different contact patch shapes though) Yes, sidewall stiffness will have some effect but it is a secondary effect for pressures above 30psi or so. As tire pressure decreases the sidewall plays a bigger part and I'm sure it will throw the formula off, especially on low pressue flotation type tires.
I've looked at many tire road test comparisons, and when the show contact patch area for the various tires on a given car, they are very similar regardless of tire aspect ratio.
I've also used this formula to figure out axle load and been within 10% pretty consistently calculating axle load by using graph paper under a tire to measure the contact patch area and knowing tire pressure. then measuring the load on a scale later.
It has worked on 50 series car tires all the way up through 80 series light truck tires running 80psi.
so I've concluded it is at least for me a pretty good first approximation and verified to me that contact patch area for reasonable tire pressure is dominated by tire pressure, not tire aspect ratio.