Popular horsemen don't say the opposite. Most people just stop listening after the first lesson. You move the hindquarters, you move the forequarters. You must control all of the feet - the hindquarters, the shoulder, the ribcage.
What happens if all you do is move the front, then the horse moves his front away and swings his hindquarters to kick you?
What happens if all you do is move the hind, then the horse swings his butt away and rears up on you?
I actually really like an exercise Buck Brannaman does that I use on my yearling. You disengage the hindquarters, and in the same motion, pivot away the front. Then you disengage the hindquarters, pivot away the front, and continue on in a fluid movement until you can travel the whole arena that way.