The Mongolian ponies are smaller than their western counterparts, which means the gait is such that riders can't post in the saddle as the horse trots. This results in the riders basically standing in the stirrups all day, every day. Combine this with the pole lasso that they use, and it requires an incredible amount of leg, back, and grip strength to be a traditional Mongolian cowboy.
Nowadays, they use motorcycles more than horses, and operate on fixed ranches moreso than the nomadic life of days past. Mongolian wrestling was not really a combat art-- horsemanship and archery were much more important for that. The wrestling games served as a way for young men to test themselves and gain status in the tribes. At the large games where a bunch of tribes came together, the wrestlers who showed well became the object of affection for the young ladies of the tribes, and shenanigans were encouraged as it was a way to get genetic diversity and strong children within the different groups.
My understanding is that the nomadic days of Mongolia are largely over. It remains to be seen how that will affect the traditional dominance of Mongolian wrestlers as the younger generations grow up in a softer environment.