It depends on your definition of "succeed". For most people to "succeed" at something is not simply to do it, but to do it well.
Yes any horse can do dressage, and can be trained to do the movements - usually in the form of "tricks". The video shows a well trained horse - but it does have a lot of trouble with the piaffe which is very common in horses not bred for dressage- they simply do not have the hind end conformation. I know a grade pony that did the GP but had a piaffe much like this Appaloosa's, walking behind.
Now to succeed in dressage you really do need the right conformation, the right mind and the right ability from the horse. To find a horse with these three things in combination and in the right amounts to have the potential to do Grand Prix is rare, even within the warmblood breeds, which is what makes them so expensive.
I am not saying other breeds cannot do the Grand Prix, just that it is extremely difficult that will find one which will succeed at it, even if you are purchasing a warmblood.