Show workings please or I'll assume Andrew, that you have plucked this figure from an orifice where the sun doesn't shine.
You are assuming that the tyre's material is elastic in that complete range, which I assert is most unlikely. This latter observation is (agreeing with you - no such material) nugatory (and the deduction you make from this erroneous quantatitive approach is entirely valid), but let's stay within science's bounds.
The circumference of a 700c tyre is about 210 cm, and an inch is 2.5 cm. 2.5/210 = 0.0119, which in my book is "about 1%".
Not difficult is it?
In fact, that's an overestimate, as the tyre sidewalls will bulge outwards where the contact patch is pressed inwards, offsetting the loss of volume to some extent. The cross section of an undeformed tyre is circular, and a circle encloses the maximum area for a given perimeter, so any deformation involves a loss of volume, but it's not as much as it would be if the rest of the tyre stayed the same shape.
I allowed elastic over the full range because that's the best case.
Any deviation from "elastic over the full range" is most likely to either being a sudden increase in resistance over what a simple elastic proportionality would give, or it's going to involve a permanent or near permanent deformation or failure.
Bear in mind that the energy spent compressing the tyre tread has to be recovered somehow, otherwise the rolling resistance will be horrendous. I'd contend that that means that only elasitic materials will be usable.
Unless, of course, you have any suggestions?