If we think about inertia again, this might start to seem obvious.
For maximum easy power, you'd want to use the legs to get the whole system moving (including the shoulder and arm) and only then, when you've stopped accelerating with the legs, add a bit more with the arm.
Imagine you're sitting in a Ferrari trying to throw a ball. If you're traveling at a steady 60mph, and throwing at 30mph, the ball will come out at 90mph. Nice and simple. But if you're accelerating hard from 40 to 80mph, so that you feel like you're being thrown back into the seat, and you throw the ball just as you hit 60mph, the same muscular effort as before is not going to add +30mph. You'll need to overcome the inertia that your arm is experiencing, which will cost you speed. The ball won't come out at 90mph.
So it makes sense, in putting, to use the legs to get moving and then, while moving (but no longer accelerating), use the arm.
Of course, there's lots of different styles and some people don't use the legs at all. But accelerating with the legs at the same time as the arm is generally pretty inefficient - you want that lag.