Re: "natural" or "evolutionary movement"
In that first video when he starts walking, he's describing it as "push, push, push, push." Meaning that he's engaging large muscle groups in the legs, meaning that he's not discharging the energy from his legs.
With core movement, the leg muscles don't do the moving; the core does. Rather than "push, push" with a weighted-loaded back leg, try just swinging an empty back leg - that's been discharged of energy - into the place the movement of the core directs it. Instead of "push, push," It should feel more like, "release, release, release, release." And then "swing, swing, swing, swing."
You should feel no sense of weight in the legs as they move. If they feel weighty, they still have energy in them. We want to discharge that energy - allowing it to drain out of the leg - before we swing it - effortlessly - in the direction determined by the movement of core.