Maybe it's my pseudo-math background talking, but I find them relying heavily on a concept of center mass to be a cheating. They basically make very convenient assumptions about the CM that, while intuitive, still need a proof of correctness.
Center of mass is directly related to the centroid or first moment of area if you would like to look into it more. It is based on integration of a product which is why simple summation of the product for a discrete element is also valid.
I kind of agree with you here. The writeup is not clear about why the CM is so important.
They could use a diagram of opposed forces along the interface between the block and the table showing how, if there is no force applied to both the left and right sides of the CM, there is no way to get the torques to sum to zero. That would connect their assertions to the fundamental equations of state.
