I think you will not be finding all the actual stylized dressage movements happening in an untrained horse at liberty in a recognizable fashion that would score you points. I think that what the old trainers meant is that dressage movements build on ways the horse naturally uses himself. As one of the posters said, if you think about half-pass being built on the foundation of haunches-in, then you will see horses at liberty doing movements with their haunches in toward the center of the circle.
The point of the claim is, I think, to stress that dressage and especially haute ecole builds on natural capabilities of the horse, and is not a circus trick or a perversion of the horse’s real way of moving, or cruel, but a development and refinement.
That said, horses improve their way of going at liberty as their training progresses. My mare has improved her trot, and somehow learned to do barrel racing circles at liberty, though we’ve never worked on anything like that under saddle. And she will passage for about 30 seconds when she gets turned into pasture with her old friends for summer vacation!
I was going to suggest her playing fetch with a glove and happily climbing on a circus box as examples of circus tricks that are unnatural, but indeed I taught her those things by building on her natural behavior (grabbing towels off the stall door hooks to get my attention). So maybe you can’t really teach a horse anything that isn’t already in its repertoire?