The winning horse at the last show I was at was an APHA. He was supposedly schooling PSG. He wasn't your typical build for a paint, could have passed for a small WB.
Typically stock breeds just aren't built right to handle advanced dressage movements, but that doesn't mean they aren't out there. I think there was a post on here about a foundation app competing Grand Prix.
If your horse says no, you either asked the wrong question or asked the question wrong
And God took a handful of southerly wind, blew His breath over it and created the horse