It might be that one legged balance doesn't improve 'balance' and so doesn't improve your health outcomes. Or 'balance' is irrelevant, and it's just that 'fitness' means you're less winded at the top of the steps, and so you're less likely to fall down at the top.
I'm gunna need a citation on that one. Training you balance is likely to increase your overall balance. Like, by definition. If you can stand on one leg well, then you have the muscles and the equilibrium to balance yourself. Therefore you are less likely to fall.
I'm saying that this experiment:
1. Does not define 'balance'
2. Does not show that the test they applied predicts or correlates with 'balance'
3. Does not show that 'balance' causes good health outcomes.
Perhaps 50 years of dance/yoga/gymnastics/core strength/martial arts leads to both fewer falls and greater 1-leg stance duration. A much shorter course of balance training much later in life might be insufficient to replicate the same benefits.