A comment from a non-competitive beginner...
I started to use the correct kicking very early in my drills. So when I did a right hip drive I would start it with a kick from the left food. In the drills I am flutter kicking in between switches, but switches I always did with the correct foot kicking to initiate the hip drive.
The effect was when I started with full strokes it was rather easy to get into a 2BK. The only thing that prevents a nice 2BK is balance problems so I flutter-kick sometimes in between and my legs sometimes do strange things to counteract loss of balance. But to use the right (=not wrong) foot at the right time is almost automatic.
The other day I felt a little high-spirited (I don't know if this is the right word...) and I wanted to do a fast lap with kicking. So I figured it out and thought, ok, the only way is kickwithhipdrive - kick - kick - kickwithhipdrive - kick - kick - ... as Coach Dave said. I started the lap and it worked from the very beginning, although I never drilled a 6BK. The kicking as the initiation of the hip drive was so imprinted that it worked on its own and I was actually kicking a nice 6BK with my strokes although I not even once drilled that. Probably it was very slow but I was amazed how much propulsion came from the kicking and I felt like a torpedo :)
In short, to imprint the kick with the hip-drive might be enough to be able to swim with 2BK and 6BK. Yes, I admit - I am very fond of drilling. I know, not everybody is. It's boring and not cool. But it brings so much benefit.