Bottom bracket height appears to have very little to do with stability as far as bicycle dynamics goes.  Unlike cars, where a low CG is important in managing body roll and weight transfer, bicycles must lean to turn.  A lower CG makes a bike more stable in the same way a low CG makes a car more stable, but a higher CG also gives a bicycle a larger moment of inertia when leaning, which actually makes the bike more stable (i.e. harder to lean) because with a higher CG, you have to move the same amount of weight further to lean the same amount.  They don't quite cancel each other out, but the narrow range of BB heights relative to CG height makes it not matter a whole lot.

If you want to see this effect in action yourself, try balancing a baseball bat or broom on end in the palm of your hand and compare it to ballancing a pen on end in the palm of your hand.  Or try out a tall bike.  Aside from the difficulty of putting a foot down, they're much easier to ride than they look.