The tyre shape, profile, layup, material, different rubber mixes/densities etc. all contribute to the overall handling including the damping. A modern tyre is far from just being an air container.With an airless setup you have more control over spring rate and damping
Also, as far as the whole idea that air tyres rebound perfectly - I run mousse inserts in my rally motorcycle (I'm pretty sure jbjork also races his bike on mousses most of the time). The tyres do feel a bit more "dead", but whether this more to higher mass, higher damping or some other interaction, I'm not sure. Frankly, I prefer the feel of the air, but it's too risky for high speed offroad racing.