I have read of people dividing mature shrubs with a saw and planting them all successfully, but I do not remember any details. Do you?

The reason why it's not accepted to be correct is because plants will abort top growth by themselves and pour their energy into root formation. It's a survival tactic sort of like when humans get too cold. We divert our blood (warmth) to our core body, and if the temperature keeps on dropping, we divert out blood into the brain. Excuse the comparison, but if humans were to 'freeze' our extremities would be a 'small sacracrifice' for lack of a better comparison compared to saving the core body/brain. In plant, you will notice that some top growth will die. In the case of the plant, the 'core' would be the roots (which are also capable of producing new top growth). The plant will need every possible photosynthesizing leaf to create food, and those that the plant doesn't need will wilt and die.

Second, when you cut/trim top growth, you no longer have one growing point, but two. Remember there are 'dormant' or inactive buds at the nodes. Once the top growth is cut, the energy normally sent to the single top shoot will be diverted (not always) equally to the dormant buds. That's why when we pinch annuals for a bushier plant. It actually takes the plant less energy to support one central shoot rather than a number of smaller breaking buds.

But do hear, it's not that pruning in proportion to root loss is totally wrong, because it was the accepted theory for quite some time. It's just that we keep on discovering better methods and techniques that the older accepted theories just fall out of place.