I live currently in Tennessee Walking-Horse country, and watching them do their little goose-step thing is seeing one of the stupidest things I've ever seen animal do. I admit I have no idea the history/context, but I live 20 miles away from the biggest "Walking horse celebration" festival in the country, and you'd have to pay me over $100 to get me to go to it.
That high stepping with the front legs is not a natural thing, people put weighted rings on their front legs to irritate them so they learn to step like that. There are a few breeds that people do that with, I forget the name of the Spanish horses that they also do that to. They used to permanently attach the metal rings, now they use removable ones, but big deal, you are still basically hurting the horse just to make him do that stupid step.
The running walk gate is not that high step thing, it's a natural way certain breeds move their feet while running that makes them smoother to ride than other breeds. Instead of rocking forward and back you hardly move at all in the saddle. The Peruvian Paso Fino horse has a similar running gate.
Strangest ride is the horses they use for harness racing, there's a few of them at the ranch that have come from the track, they are trained to move in a way that when you start running you are bouncing side to side, very hard to untrain them from that and get them doing a normal lope that has you rocking forward and back.