I would certainly agree that the mechanical properties of tyres are complex & difficult to quantify reliably, riff_raff. A couple of comments on your post, if I may.
You mention sidewall hoop stress as contributing to sidewall stiffness. I would suggest that sidewall shear modulus & stress also contribute, probably significantly in some cases, if not all.
You also mention additional tensile forces caused by rotation. I agree, but how that is split between the equivalent of "preload" and a genuine increase in stiffness is hard to assess, I think. I have seen a model of tyre stiffness compiled (from track data, I believe) by a group who should know, & this did include a V^2 term. It also assumed that the pure load-deflection slope was constant. Now I can disprove the second (tyres have a "rising rate" characteristic), & that raises the question in my mind whether the V^2 term was caused by rotational speed, or because downforce (& hence mean load) also increases proportional to V^2 (track data, don't forget).
In fact, I think it is necessary to consider carefully the definition of stiffness. I executed a series of tests on a single, non-rotating, tyre a few years ago in an attempt to improve my understanding of tyre properties. The tyre was mounted on a wheel located by a stiff spindle mounted in a modified "dyno" frame. The tyre was loaded by an hydraulic actuator with good position & velocity control (position, as observed by a concentrically installed LVDT, was held with an accuracy of around 0.02 mm). I was as careful as I could be about maintaining constant tyre pressure & temperature, & each run was preceded by a stabilizing period to ensure that load/deflection was consistent.
Here is a plot of a selection of results from the tests. Run 331 was recorded whilst the actuator described a slow (20 second period) sine wave. The regression slope was 345 N/mm, and the rising rate characteristic is obvious, I think. Runs 326, 329 & 332 were recorded whilst the actuator described relatively fast (5 Hz) sine waves of constant amplitude, each run having a different mean load. The regression slopes were, respectively, 334, 395 & 408 N/mm. (again showing a rising rate characteristic). Now, if the three high frequency runs are thought of as three steady state conditions with superimposed "dither", then the mean values of the three sets of measurements (shown as yellow circles) can be used to obtain yet another estimate of "stiffness". These lie close to a straight line (shown in red) with a slope of 289 N/mm.
The obvious question, I suppose, is what is the vertical stiffness of this particular tyre? I suspect that if you were interested in the average deflection of the tyre at a given mean load you would choose one value, but if you were interested in the way the tyre interacts locally with the suspension (as I am, usually), you might well choose another....
