Is there something in the BOSU ball that I'm not aware of? Is it filled with a slurry of emu oil and electromagnetically charged ground up unicorn horns?
I think the far majority of exercises on the BOSU are simply, well...dumb. At best they are a waste of time relative to other aspects of performance and fitness that you could be working on during training. For the most part, they are a pointless test of your ability to remain uninjured.
Here's a case of high injury risk with little, if any benefit. Tuck jumps demand a lot of the legs and core. When many athletes have difficulty jumping and landing on flat ground with adequate control, this guy is having his clients do it on a BOSU.
Truly, there are so many dumb things you can do on a BOSU.
These are not going to improve your strength, stability, or mobility any more or less than pretty much any activity that does not involve you sitting in front of a TV or computer. These are not even going to improve your balance in a way that is relevant to functional athletic performance. The one thing that such BOSU exercises achieve is a fairly fast and drastic improvement in performing that BOSU exercise.
If you're going to load your muscles, bones, and joints with resistance, do it in a way that encourages your brain to practice proper control through a full range of motion. When that gets easy, make the exercise more difficult (and beneficial) by gradually increasing the load. The far majority of athletes desperately need to improve their functional mobility (stability and range of motion) on solid ground.
Whether your primary goal is weight loss, conditioning, strength gain, or what-have-you, most of us will stand to benefit the most from plain GETTING STRONGER, something that the BOSU inhibits relative to training on the flats.
Single leg squat on a BOSU? Single arm plank for time? I promise that you can do almost any of those ridiculous looking BOSU moves and poses if you have good strength and you practice that move.
I will grant that balance training on the BOSU and other unstable surfaces is indeed beneficial in the context of physical rehabilitation. The BOSU is a decent tool for when your primary interest is sending proprioceptive input to the brain. Here are a few examples of appropriate use of the BOSU.
But please understand that there is a relatively narrow window here. Once you establish good control of these type of movements, you're far better off gradually increasing the speed and/or resistance of movement on flat ground.
[I'm not listing references here, but certainly let me know if you want to see evidence for some of these claims. For the most part, they all say that unstable surface balance training is modestly helpful for rehab after injury or surgery but offers very little benefit in the area of sport performance].