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.
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.