Many of the warmblood breeds have been specifically bred for dressage.
In a dressage horse destined for the FEI levels, we want to see a horse that will swing it's back, and be able to 'sit' behind. Therefore the angles in the hind limbs need to be quite great, with a fairly low set tail, fairly short cannons and a strong croup. A horse built uphill is looked upon favourably, as generally it will find the collected work easier than a horse built downhill.
The shoulder should be well sloped, which allows for greater knee action and swinging paces. The neck should come up out of the shoulder, set quite high from the wither.
Iberian breeds, such as the andalusian, are generally built as described above. They tend to find the collected work very easy, however, they are not generally bred to extend their paces as well as collect, which is their downfall in competitive dressage, vs the purpose bred warmblood that has been bred to show quality in both the collected and extended gaits.
Here is one of the current top international grand prix combinations warming up before a competition
These are two warmblood, dressage 'supersires', Donnerhall and Gribaldi
And here we have two quarterhorse stallions (I know nothing of QH lines and breeding so have selected the first two that came up in google)
Can you see the difference in how these breeds are built, to suit their different purposes? Some quarters horses or standardbreds may have nice enough paces, their legs might like pretty enough when they're moving, but because of how they are built, it is very difficult for them to reach the higher levels of dressage. This is not breed prejudice, it is just fact.
Just like a hanoverian bred for dressage is not going to excel in reining when pitted against a reining bred QH.
Basic dressage training, is not necessarily dressage. It is good, basic riding. Having the horse go forward, off the aids, softly into the bridle and being able to move laterally off the leg. This is very very very basic training that forms the foundation of many disciplines, so I do not claim this as 'dressage'. This basic riding is great for any horse.
It is when you begin to demand collection, that the horse's build and talent comes into play and where most candidates will get knocked off their ledge.