That video was rather confusing. Most of the clips (well, all of them, from what I could see through the editing), were complete accidents, of horses spooking or misbehaving. The riders all seemed unprepared and did not look like they asked the horse to rear. All horses rear. It isn't actually something that has to be taught. All horses buck. Again, it isn't something that has to be taught.
Stallions rear at each other to challenge. My mare does that as well.
Yes, you can get squished. You can also get squished when a horse bolts and slips, or falls over a fence, or trips on concrete, or knocks you over when he sees an umbrella.
A well-balanced rear is pretty safe. Did anyone ever stop to think that a conditioned, well-trained, well-balanced horse who has been taught to use his body correctly, probably is much safer to be on if he ever decides to rear, than some crappy, ill-trained, unmuscled horse with zero balance? If you horse can do a correctly balanced canter in self-carriage, I think he has a better shot of staying upright than the beast who's currently 4-beating around the arena cause he has no idea how to use his hindquarters.
Anywho, the rider often contributes to accidents as well. A lot of riders get taken by surprise and lean back when the horse rears, even pulling on the reins. Yeah, um, you just threw 150lbs backwards and then pulled. What do you think is going to happen?
I'm lucky that I automatically respond to a rear by leaning forward, following the movement. I can do it without thinking and that's awesome for me. Other people have to train for that same response.
My own horse knows how to rear on command. She does it because it's fun. However, she has been trained not to do it unless I ask. She's also not used as a riding horse, so there's no danger of a rider getting squished anyway.