There are two "modes" of wheel stiffness. There is firstly the small-deflection mode, where the elasticity of the spoke's net load path on both sides of the wheel contribute to the rim's lateral support. Note that the "elasticity of the spoke's net load path" includes the spoke count variable as well as the spoke's dimensions and head configuration (straight or bent) and the hub flange structure as well.

There is also a higher degree of rim loading and deflection where some of the spokes on one side begin to lose all tension, where the stiffness of the wheel is seen to suddenly drop off, and where higher initial spoke tension (pre-load) can extend this deflection point of the rim where that onset of zero tension and reduced rim stability begins.
But in both modes, at any degree of rim loading, bigger spoke diameter contributes to increased wheel stiffness as seen at the rim, and vice-versa.