I agree with anebel here. Depends on what you mean by "succeed." For me personally, it's doing a canter half-pass that doesn't completely suck. :) No designs on FEI. But I know what I'm looking at. :) Any horse with decent conformation and more importantly, good training, can do dressage up to a point. Quarter horses, appys, cobs, fjords, whatever. The appy in the video could do most of the GP movements correctly, though the canter pirouettes looked a bit flat and laboured and she did the "walky" piaffe thing. That's awesome. If I could get my horse to do even a rubbish piaffe, that would be an amazing success. It's more than most of us will get out of our horses.
BUT to be competitive in international GP dressage, you need something that can do all the movements really correctly and better yet, with a bit of flair. Think of horses like Rembrandt, Bonfire, Brentina, and Blue Horse Matinee. They need the correct conformation and the correct temperament to accept that kind of intense training and have the physical capabilities for those movements. Even then, there are always compromises. Few warmbloods have the piaffe, passage, and canter pirouettes you see in Iberian breeds, which are close coupled and have incredibly powerful hindquarters, and loads of high action. But the Iberian horses, for the most part, don't have the spectacular extensions of the warmbloods. Not many Iberian horses win at international dressage (I saw a few Andalusians and Lusitanos in the last summer Olympics, but they got soundly beat by the WBs. Those big extensions count for a lot). But not many warmbloods are doing haute ecole. They're not built for that extreme collection.
The point is that for us amateurs who are happy doing dressage at low-ish levels, no, a super fancy horse isn't necessary. You can do correct movements, provided the training is there, but there is a point in dressage progression where the correct execution of certain movements (i.e. Piaffe) require the horse to have physical capabilities a lot of horses simply don't have.