It's always interesting to see a technique evolve independently in multiple domains.

I practise taijiquan (tai chi) which is very heavily predicated on this principle. A huge part of it boils down to practising being stable on one leg so the other leg can move anywhere. You might be interested in the taiji stepping techniques, but these are not (visibly) part of the forms people generally think of as tai chi, and not all schools bother to teach them.

Of course, it's not the same thing TFA talks about. Especially in the context of traditional Chinese martial arts, with all their associated mythology, I think it's helpful to remember that human physiology is pretty unique, especially when it comes to balance. We're inherently unstable, so becoming more stable takes practise and conditioning. We're not flamingos (or snakes, or white cranes).

You said yourself that putting the second foot down doesn't increase stability. It's a paradox.

To answer this from the taijiquan perspective, the stability comes from being properly balanced on one leg. The second leg is a spare for when something happens unexpectedly like a gust of wind. By forcing people to keep one leg off the ground, you're forcing them to balance on one leg and learn to be more stable like that.

If you have no balance, it doesn't work. That's why tai chi is often recommended for older people to improve their balance and reduce the risk of falls.