The answer can be best summed up by answering these questions.
Can you imagine a person doing a bhs with no perceptible arch?
Can you imagine a person doing a bhs with-out tipping their head back?
Can you imagine a person doing a bhs with out reaching around over their head with their arms?
Can you imagine a person doing a bhs with out pushing with their legs?
If your answers were yes, yes, yes, and nooooooooo, you got it right. Sadly the kids we teach have a hard time concentrating on the leg push because they think the first three are so important, and they destroy their ability to use their legs by getting into the most human hostile positions imaginable.
A good strategy is to understand that their mental model of the skill is way out of wack, and to make consistent efforts to change their mental model to one that works. That model would favor the leg push over everything else, because done right it's enough to make all of the major parts, flight, momentum/energy, and tipping upside down.
If you're doubting the upside down part, consider what happens to a broom stick you're holding at vertical balance in the palm of your hand. If you push up just right on the end of the broom stick as it looses it's balance it will turn end over end....just like a bhs. So a bhs needs to start with plenty of energy from the leg push.
You could use a jump back drill onto a resi, or depending on your equipment and ability to foresee accidents...an octagon and into the pit to experience the "flip". I'm sure there are plenty other drills people have to offer, so choose a few that make sense to you......and as long as they don't de-emphasize the leg push, or interfere with it, you'll be good to go.