To be fair to Buck, that quote isn't too far off Joep.
You do use the left leg to "kick" with the right leg.
There's a "stretch" component that allows for this kind of movement. It's not immediately obvious, takes some conditioning, but once you get it, you would never think of going back to the other way of kicking.
Agreed, but when I read this
We lift the top of our head up, make our front foot gently push into ground and raise our body slightly. That way our back foot is lifted off the ground.[emphasis mine]
I see someone bobbing up and down while walking, because to raise his foot off the ground, he keeps his body rigid and extends the supporting leg. Then he moves the foot forwards and puts it on the ground by bending the supporting (back) leg. Not a good way to step, in my opinion.
A better description would be: "We lift the top of our head up, make our front foot gently push into ground [that's up-down contradictory tension and groundpath if you're generous] and draw the back foot upwards and inwards
Apart from the bobbing issue, it becomes clear that lifting the foot is not really different from moving the foot forwards. (And thus there is no need to talk about moving the foot with the legs and hips, which is the only way to move your foot anyhow.)
So perhaps the quote is not too far off, but it's far enough off to lead you in the wrong direction. Especially since it doesn't mention the conditioning that is needed. If nobody tells you about the 'stretch', you probably won't discover it and end up doing it wrong.