Stabilizer recruitment is also load dependent, not just movement pattern dependent, people in the functional training world always seem to miss that point. This means that the stabilizers have to work ten times harder doing a 500lb squat than they will ever work doing a 135lb squat on a bosu ball. What is actually happening in unstable surface training is the ankle joint is getting a great workout and the body is learning the muscular coordination to not fall over, other than that there is not enough overload to develop any real strength.
A few years ago a PhD student friend of mine did some research with JC Santana because JC believed that if you added up all the force of each joint in a single arm cable pulley press it would actually be higher than the force generated during a standard bench press. This is the argument the functional training crowd uses a lot of the time to justify unstable surface training, that more muscle is actually recruited because of the instabilty. The result of the research was that a standing one arm cable pulley press produces less than 1/6 of the total force generated during a standard bench press!
These types of functional training methods completely miss the point of what specificity really means and do not generally provide enough overload to cause any real adaptation. The only thing that really happens is the athletes learns the motor program and is more coordinated in that specific movement, but there is very little to no transfer.