Any horse can compete in dressage.
Because of the movements required in the upper levels many individuals(including fresians, warmbloods, saddlebreds, and quarter horses (etc.) never make it to the top levels.
I think each breed's conformation will either allow the horse to master the levels with ease, or make it quite difficult for the movements to be performed.
If you were to take a QH that fit the breed confo. Specs perfectly, and a hanoverian that did the same, start them both with the same trainer at the same time, and check in on them in 3 years, the warmblood would be doing the movements more naturally than the QH - - It's just how they are built. The QH would be able to perform the movements, but it would be harder for him.
By the same token, if you were to take another "perfect" QH and another "perfect" hanoverian, and teach them both to do reining, well, the QH would be performing the gaits more naturally - - just because that's what he's built to do
So long story short, any breed can do dressage. But if you want to move up through the levels and not cause undo stress on joints/etc. in the upper levels, then why not go ahead and get something that is built to do it?
By the same token, the first few levels really don't require that much collection or stressful movements, so if that is your goal, any breed can do those levels