The Nooitie can certainly compete in all these disciplines, but at this stage they can't really compete with sport horses and warblood types. They are relatively small in size - min height is 13hh and max height is 16hh, but the average Nooitie is about 14 - 15hh. My stallion is 15hh and my gelding is quite big at 16hh. Originally they were used as farm horses, working sheep and cattle, having to work long, hard hours under the saddle. But, they were also the reason why the Boere won the British back in the Anlo Boer war, because they knew how to survive in the veld without extra food, and carrying the quite heavy Boere for many hours. They went barefoot and had a very special relationship with their owners, and therefore they lasted, while the British troups rode imported horses who couldn't survive in the veld without extra food which they had to carry on their backs, they went lame over the rough terrain, and generally couldn't keep up the hard pace and long hours under saddle in unknown territory.
So yes, while Nooities are used extensively in the children's market in all these disciplines, they are not used so much by very competitive adults, because of their size. But, what they lack in size, they make up for with their huge hearts. People owning Nooities can't stop talking about their special relationships with their horses.
Here's a stallion that did very well in Hunter show classes and even won the warmbloods and THB in a Supreme Championship once. His name is Pretman Tornado.
I'm just trying Western Pleasure and Western Horsemanship with my boy at the moment. As I said, WR is still very small here in my part of the world, so we do it on very small scale. There are a few competitions that I can attend, but it's not always possible to go with out African Horse Sickness regulations and far distances to cover. But I do go to the ones I can.