Reading the start of this felt like getting nerd sniped.
While the Horse did win the competition of "Which animal is the best for riding, and pulling carriages?" they did not win immediantly.
There are many examples of people riding non-horse animals, including two attempts at Military Moose Cavalry!
People in ruraler areas with less access to horses, both from location and a lack of money, used every other hoofed animal of sufficient size. Cows, sheep, donkeys, goats, elk, deer, buffalo, moose, camels, pigs, elephants are all known examples.
If it was at hand, large enough, and tame enough, someone used it.
If you google 'riding elk' you get a photo from 1910 Montana of a man riding an elk in town.