I don't see why they bother to use a western saddle. Western saddles are functional. They are made as a tool for ranch work. Whether for working cattle or to ride while making repairs. Otherwise they are unnecessary.
Riding with one hand on a loose rein and getting the same, and better, movements with the horse like shown on the videos is also functional. Riding the western horse and working you need one hand free. The horse is a partner, doing it's part, and with little input from the rider at times, so the rider can do what s/he needs to.
So I won't say one is better, but perhaps one is unnecessary.