Dressage is a tried and true method of training a young horse through certain exersises and movements to be better able, physically and mentally, to perform in any equestrian sport.
Thinking about different equestrian sports, very few are actually real life anymore. A highest level dressage horse is pretty to look at but there is no real use for an extended trot, for example. A reining horse does spectacular stops and spins, but working a cow, a stop like that would send the cow off back to the herd, laughing.
But still, to perform these somewhat exaggerated movements, a certain level of rideability is needed. And this is described well in countless books about riding.
To achieve this level, any horse is best put through schooling certain figures, maneuvers and exercises. The only "school" who has a general standard, tried and true, since many years, happens to be dressage. To make that clear, dressage as SCHOOL, not as competitive sport.
So why shouldn't the average westernrider benefit from this kind of schooling?
I don't think there is any danger that one of them tries to take honor and glory away from a grand prix horse. Or a champion reiner or cutter.
Oh, and the cutter trained to 3rd level dressage has most likely no intention to compete in dressage, but is still capable of performing 3rd level exercises. And I bet he wouldn't ask a 3rd level dressage horse to cut a cow or two.......