Math theorem: the most misshapen ham sandwich can always be cut into two perfect halves

No, that is not correct. The centroid is the effective center of volume, but it is not the case that every plane through it bisects the volume. Consider, for example, a single object consisting of two spheres of unequal size separated by a long line segment. The centroid will be on that segment between the two spheres, and a plane through that centroid orthogonal to the line will not bisect the volumes.