Ok, I made a mistake in my assumption, and then compounded it by trying to extend a 2 dimensional model to a complex multidimensional reality. For auto tires, it turns out that there is a derivation and experimental data to back it up (quoted from an engineering forum: http://eng-tips.com/viewthread.cfm?qid=94153):


"Distance traveled is somewhere in between what the free radius and the loaded radius predict. There's a derivation given in "Mechanics of Pneumatic Tires", that identifies the tread as being compressed in the contact patch and also over the zones immediately before and after the contact patch. It goes on to talk in terms of effective radius and effective deflection that differ from the loaded radius and actual deflection respectively. And it mentions that very little longitudinal slip occurs within the contact patch (assuming no acceleration/braking), hence there's little wear under conditions of rolling in a straight line.

Eventually the discussion gets around to providing some experimental data, with the actual distance traveled measured for a bias tire given at 96% of what the free radius predicts but the loaded radius being only 94% of the free radius (yes, this dates my reference material somewhat). For radials, it gives 98% distance traveled with the loaded radius being only 92% of loaded radius. Data for more recent tires may differ in the specific percentages, but I'd certainly expect the general relation to hold."

Assuming the data for bicycle tires would show a similar result, then my simplistic model doesn't work well in the real world, and it looks like measuring rollout is the only solution - with a loaded measurement giving the best result. mea culpa.

JB

edit: while I'm eating crow, I should also point out that I did make an error with the geometry in my 2d model. I'd assumed that the deflection acts like a secant across the circle, but that the circumference would stay the same. The only way this could work is if the radius of the circular part of the deflected tire has a radius LARGER than the original, unloaded radius. Obviously, that completely invalidates my assumption of no stretching or compression going on - the tire would have to stretch along the circumference to account for the larger radius, but then compress along the flat part to achieve the same overall circumference. mea culpa, mea culpa. I'm going to go enjoy my crow dinner now (and thanks to my son's geometry teacher for pointing out that error!).