Actually, that was fairly close to how things WERE done. See the pictures below from around 1900-1910:
All pictures from Erwin E. Smith Collection Guide | Collection Guide
The Erwin E. Smith Collection is the best collection of photographs I've seen of working cowboys from that era. It is well worth browsing - but you will see a lot of high held hands, high heads, etc.
Now, they didn't race up and suddenly stop and dismount mid-stop just for fun. THAT is Hollywood! And much of the time on a cow pony was walking:
But they were not dressage riders, and many were not interested in fine riding. Most cowboys of the day were young. They had a hard life, and their animals shared it. Teddy Roosevelt, writing about ranching in the 1880s, distinguished between horses you owned personally and broke for your use, and ranch horses supplied to the cowboys - that were broken in 3-4 rides in a few days by professionals.
In their defense, I doubt they were all that abusive of the mouth. There was a lot of slack rein time. But when they wanted to stop, the horse had better stop. And given what could happen if a horse didn't stop, I don't blame them.
Also, the dressage idea of collection isn't really the end all of riding. I've never seen any empirical data showing that collected gaits make a horse live longer or function better. Nor is a dressage headset a good idea covering rough ground at speed.
It might be that we know better now, but it also might be that the riding style they used was appropriate for the type horses they had and the work conditions they faced. I've never tried to push 2,000 steers thru unknown country for hundreds of miles. I've never ridden a half-broke horse for 24 hours straight in pee-poor weather, knowing that a fall could kill me. I'd be very careful before assuming we know vastly more than they did then. In some areas (jumping comes to mind), we DO know a lot more than a rider in the 1860s. But what most of us do not know is what the riders of the time really faced, day-to-day.