Nearly all of the research in this search says there is either no statistical difference in effectiveness or “The diminished force output suggests that the overload stresses required for strength training necessitate the inclusion of resistance training on stable surfaces.”
The research doesn’t go counter at to any weight training principles (nor it should at all). I would definitely have my clients training on a stable surface to maximize their strength gains (due to a higher cost of failure if done on an unstable surface).
Every piece of equipment has its time and place and any good trainer should recognize the strengths and weaknesses of each type of equipments. I guess the unfortunate matter is when a trainer or any person working out gets obsessed with a particular type of equipment that they want to apply that to everything they do. This is kind of how “functional training” (which should actually be simple to get anyone to do) becomes something of a circus act. Trainers or uneducated people adding in equipments to make things a little bit harder and justifying it as “functional training.”
The SAID (specific adaptation to imposed demand) still applies, and if my goal were to make somebody as strong as possible in any given exercise, I would not opt for an unstable surface. if I were sure that the person already had very strong core muscles.
However, in the process of getting him/her there, I like unstable surfaces and usually see improvement in peripheral strength just by strengthening the core. Personally, I like to include unstable apparatus for it but Pilates and other modalities are just as valid.