Boxers, wrestlers and MMA fighters do all kinds of exercises for stabilizers. Farmer's walk, pushing sleds, lifting sand bags, plyometrics, calisthenics, etc, etc. No fighter trains for strength exclusively on machines and most never use weight machines at all.
Are you able to point out specific muscles that are "stabilizers" as oppose to just skeletal muscles? Is there a specific class of muscles classified as stabilizers? To me, all muscles in a sense are stabilizers in that they hold your body together along with the skeleton. But if all muscles are stabilizers then it is a meaningless term. It's just muscle.
The argument I hear is that because free weights require "stabilizing" muscles they are harder to do and therefore more productive. So the implication being is that the more unstable the environment the more productive the exercise.
If I want to target my quads, it seems to me that the more unstable the environment the less I am able to target my quads because other muscles are taking over to maintain this stable environment. If recruiting more stabilizing muscles is a good thing to stimulate hypertrophy on a targetted muscle, shouldn't squatting on a Swiss ball be more productive than typical barbell squats? Certainly, squatting on a Swiss ball is much more difficult but is it more productive? It is if your goal is to enhance your ability, your skill, in Swiss ball squatting but because all these other "stabilizing" muscles are needed to perform the movement it takes away from maximizing resistance, and therefore stimulating an adaptive response, for your quads -- again assuming that you are squatting to increase the size and strength of you quads.
Mike Mentzer once joked that if he knew how important "stabilizing" muscles were and how an unstable environment promotes this, he would have hit the gym during an earthquake instead of seeking cover.