"Most of the 'cowboys' who went out West to work on the big cattle ranches were of Irish & English origins. They were boys who'd learnt to ride before they came to the US and handed their knowledge down through the generations So their riding must have been influenced by the european style of riding that was in turn influenced by dressage..."
False. Most of them were self-taught, or taught by the military. Their style of riding reflects the style used in the US Cavalry, which rejected dressage as unsuitable for military needs - needs that include fast training for horse and rider, and covering a lot of ground with the least effort. And no, they were not mostly immigrants.
From the link you provided: "The United States Cavalry at Ft. Riley exchanged ideas and instructors with the schools in Europe and started the trend that brought dressage training not only to the military but to civilians in the United States.
Not true. The US Cavalry rejected dressage, and the manual written in the 40s - the last one written - finally mentioned collection, but was far more heavily influenced by Caprilli than by dressage. I've got a copy, and read it closely. It is a good discussion of how to jump, but it is significant that it was approved about the same time horses disappeared from normal war use.
Here is a picture of cavalry from the first World War:
Compare to a cowboy of that era (1907 Texas):
That is an awful lot like how folks were riding in 900 AD:
The style of dressage adapted that for the demands of riding a very collected horse, which is part of why the heel moved under the hip. People were not stupid about riding horses for 2000 years. They rode they way most people have used:
That is just the easiest way to ride and control a horse. It isn't dressage, and dressage doesn't mean 'horse using butt'. And it was also used by the Chinese, and by the Indians in the west. That is the trunk, not the branch.
The idea that dressage is the mother of all riding is simply not backed by any evidence. It is a myth.