Any question answered...

Ok, my mind is slightly blown

It shouldn't be if you do a simple thought experiment. When gauging wheel stiffness, you're really asking "how hard will it be to push the rim to another position with respect to the hub?" When you try to push the rim sideways, half the spokes are resisting you and the other half are pulling on your team. However, since you know the rim won't move all by itself, it must be the case that until you turned up, the two teams were deadlocked. Now imagine each team starts pulling twice as hard; still deadlock, and your contribution still tips the balance just as much. In the case of radial movement, the opposing forces are spoke tension and rim compression, but they are still balanced before you show up, and they are balanced whether the forces involved are large or small, otherwise the rim would be accelerated in the direction of the net force.