During one of the many Mongol invasions of Korea (~1237 C.E.) the Mongols destroyed many of the written records leading up to that time, so Korean history before that time is not well documented. That having been said, middle-ages Koreans did at least have
some documents that described their martial arts. In 1608 they published the
Muyejebo, "The Compendium of Several Martial Arts", the
Muyesinbo "New Compendium or Martial Arts", and of course most taekwondo buffs know of the
Muyedobotongji, "The Comprehensive Illustrated Manual of Martial Arts." As I understand it, each of these references builds-upon and adds-to its predecessor.
One would think that if
kicking people off horses was a standard Korean technique during the middle-ages, it would have appeared in one of these references. To my knowledge, it doesn't. As I understand it, what these and other references
do indicate is that knocking people off horses was indeed important to the Korean military, because of course that's how Mongols always invaded: on horseback. As I understand it, that's why the
bow became the preeminent Korean weapon of choice during the middle-ages, over even the sword, the spear, and also presumably over all the other little batons and nunchuks and other hand-weapons common in other Asian countries: if you want to knock a Mongol off his horse, a bow is apparently your best choice, and small hand weapons aren't much use.
That having been said, at least the
kicking-people-off-horses story a very colorful story that conjures some entertaining mental imagery.
I think the explanation given by other folks here is the right one though: ancient Korean martial arts apparently tended to favor kicking, so in order to better distinguish taekwondo from karate and other martial arts, and in order to make taekwondo feel distinctly Korean, the post-WWII masters tried to incorporate as much kicking as possible into their arts. I'm glad they did, because I do think it makes taekwondo interesting, unique, fun to perform, and fun to watch!
Mongol invasions of Korea - Wikipedia the free encyclopedia
Muyejebo - Wikipedia the free encyclopedia
Muyesinbo - Wikipedia the free encyclopedia
Muyedobotongji - Wikipedia the free encyclopedia and
Muye Dobo Tongji - Taekwondo Wiki