Not sure how you were taught back, but most good swim coaches are going to have you learn how to nail body position and kick b/f they add in arm action. Trying to add kick timing to the arm stroke is really tough I'm not sure I could coordinate it and swim backstroke that way. The more I think about it I venture to guess I'd be lost if I tried to swim back with arm action as the focus and then add in intricate kick timing. I think this is one reason so many adults find a 6 beat kick so difficult with freestyle. They build their stroke backwards with arm action first and then try to add leg action by fitting them into the stroke cycle.
This drill is so old I was doing it in the 70's and it hasn't changed one bit! If you use the kick to power rotation the shoulder and subsequent arm action will just happen and it becomes just like freestyle in that it's a recovery stroke.
There is a reason you don't see people emulating Ryosuke and his kick technique. He is a 1 in a million with total perfection of his body position and stroke timing grabbing enormous amounts of water. Quite simply he is a freak that can get away with it.