I like the arch thing.
I don’t know about this ingrained heel toe issue. People stand too. We certainly don’t have an issue of people standing on their toes. That leaves heels. But they’re on their heels mainly because they’re scared. They look for anything to get some help, like poles.
What is ingrained is friction under the feet. People who’ve never skated or roller bladed have no reference at all for gliding.
When people stand, they are at rest and their mass is supported skeletally, primarily through the heel. IMO, it is movement while the brain is expecting stillness that is at issue. Ever enter a moving walkway at the Airport? There is plenty of friction between your shoe and the mat. You even see this phenomenon on the magic carpet belt lifts.
There are other factors for being on your heels like when you head down a slope, you try to keep your perpendicular relationship with gravity and ski acceleration only makes the situation worse.
Also ingrained into body function is stance leg and swing leg. But it’s not like just because you stand on one leg you start swinging the other.
Analogies are just that. The bike has it’s own issues. Sit down, push one foot down, let the other shorten. And of course you lean in. How to explain balance?
Actually getting someone to commit to balancing on their outside ski isn’t going to be accomplished by analogies.
This seems relevant, the reverse bicycle. People can’t ride it 10 feet.
Pro bike rider can’t ride the reverse steering bicycle, then practices for two weeks to learn it. But then he can’t ride the normal bike.
I look at balance as the COM's relationship with the forces at play. In skiing, potential energy is supplied by the gravitational pull of hill and we have Gravitational and Centripetal forces to deal with in order make the tools (skis) work while remaining upright. To a large extent, beginner's upright balance mechanics address the force of Gravity while experts change their mechanics to take on the Centripetal turning force. This, IMO is the chasm that needs to be bridged. There is a difference between balancing ON edge (Gravity pulling) vs balancing AGAINST the edge (Centripetal pushing).
As to the bicycle analogy, like
@Noodler posted, it is not about pushing down the outside but about lifting (softening) up the inside and is the opposite of bicycling, so I make it a point....a big point when describing the need for vertical leg action.
Your point on leaning is also valid however. And it is really at the crux of the matter IMO. Mechanically, it is the difference between inclination and angulation, yes? This is why I focus on what I call the Flex Complex (ankles, knees and hips). It is how we manipulate these lower body joints into new patterns, both fore and aft and laterally depending on the dominant force at play that leads to success.