Re: An exercise that illustrates internal power
The question of why to use the center is a really, really good one. I've often wondered that myself. I can say a few things about it, but I'm not going to pretend to be qualified enough to fully answer it.
Let's say we're pushing a heavy rock. We often think that the power of the quads gets added to the power of the glutes, which gets added to the power of the back, which gets added to the power of the arms. But this isn't true. The power of muscles don't really "add together".
In a chain of muscles, you're really only as strong as the weakest muscle. If you have a couple of strong muscles and one weak one, that weak muscle is going to collapse under the force of the stronger muscles.
I used to competitively bicycle. At first, my calf muscles were really weak. When I was going hard, using 100% of my quads and glutes, they would literally collapse. They couldn't stand up to the force generated by my larger muscles. My heel would drop until the ankle joint locked out. Over time, they strengthened. But still, all they ever did was transfer the force of my quads & glutes.
Technique overcomes this in a number of ways, most notably by using angles to reduce strain on weak muscles, and then transferring force through the skeleton. But still, you really only have one or two force generating muscles, and the rest just hold the joints and transfer energy.
(And note I'm only talking about raw power generation here. Quickly accelerating a limb or a small, light objects doesn't require much force, so that type of movement can benefit from multiple muscle/ joint movement.)