All else being equal, I think it's less the volume of air (i.e. the size of the air cavity) inside the lower profile tyre than the shorter side wall being inherantly stiffer (than with a higher profile tyre) that increases response to steering input as well as increasing harshness over bumps etc.
Tyre profile (and psi and sidwall stiffness generally) affects overall spring rate because the tyre is effectively the first of two springs in series (the tyre, then the coil spring itself), but it's hard to predict what the affects of increasing tyre stiffness might be for damper rates because the tyre is an undamped spring that is typically very much stiffer than the coil spring, which is damped.
Note that in F1 cars and similar, that the tyre is the major spring in that it's usually (always?) softer than the coil and accounts for the majority of the 'suspension' compliance.



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