Re: "natural" or "evolutionary movement"
In fact, the reason he's describing a "push, push" movement is because he hasn't moved his core forward enough by the time he moves his back leg. So the core it still over the back leg. The back leg is still full of energy. That energy has to drain first - out of the bottom of the leg.
By him pushing, he's sending energy back up into his body, in the opposite direction it needs to be going for effortless core movement. Imagine an eight-lane highway with cars going in both directions all in the same lanes. That's what's going on in his back legs.
Instead, if he would allow the core to more forward about a half an inch or so - and then move his back leg, the energy - which was provided by the core - would discharge from the leg, and he would just swing an empty, weightless leg effortlessly forward.
We don't want to "lift" anything. We don't want to "push" anything.