It would help if people knew what dressage is. It isn't a discipline like jumping or reining. For the horse it's progressively first restoring it's ability to use it's body, especially the back, as nature intended, then building strength and suppleness so the horse can carry a rider with minimal damage to itself. There are some ideas that aren't found in other systems - like narrowing the horse's base of support as it gains strength and agility...
Why can't this be done with a western horse?...
Because the goal of dressage differs from the goals of western riding. The goal of dressage is for the horse to move in collected gaits, which can be defined as the horse doing the most work while covering the least ground. And that is contrary to western riding, "western pleasure" not withstanding.
Western riding reasonably should be the style of riding used on the ranches of the west. And a ranch horse doesn't need collected gaits. A big, bouncy sitting trot is NOT a western approach to a sitting trot. It is spectacular and appropriate for dressage, but not for western riding.
Western saddles do not encourage lots of motion of the horse's back, because the long bars of the saddle tree are contrary to how you would design a saddle for maximum use of the horse's back. They do work well for roping, or working cattle, or for someone who may spend 12 hours or more in the saddle.
In dressage, the horse is supposed to be heavily cued, with the rider making all the decisions. In most western riding, the horse is encouraged to make some of the decision, since a human cannot respond to a cow's movement and then direct the horse what to do - not if they are going to work cattle together.
None of this implies a western rider couldn't dabble in dressage as cross training or a way to have fun or a way to work on their horse's general physical conditioning. And frankly, that is what some of these videos look like - someone dabbling in dressage. I don't barrel race, but I've gotten some good advice from ex-barrel racers on how to get my horses to move better. But borrowing ideas on training from a barrel racer doesn't mean I'm
a barrel racer.
That is part of the problem with "western dressage". It mixes two different ways of riding without appreciating either. If dressage wanted to introduce a category where someone could compete at the lower levels using western tack, I wouldn't object. It would be clear the person is learning some fundamentals of a different sport without buying new tack, while also making it obvious that if the person wants to get serious about dressage....well, then they will need to get serious.
If the WD people would change the name, I wouldn't object. But as it is, I think they insult both western riders and dressage riders by trivializing the traditions and training of both.