They do and in fact it's relatively straightforward to show empirically on eg MNIST. The problem is that you need a much much larger network in the FCN case and thus need way more data and way more data augmentation to get a good result that isn't overfit to hell.

In the case of CNN the reason it works is that an image of an object X is still an image of object X if the X is shifted left or right. The property is translationally invariant. CNN are basically the simplest way to encode translational invariance.