A cool way to add further depth into the game would be to add various horse breeds into the game that are more efficient at performing different tasks. At a basic level you could break it into three categories, Draft Horse, Light Horse, and Warmblood Horse.
So your Draft horses would be the stronger of the three, able to carry heavy loads with ease and at a faster, steady pace.
Light Horses are your speed and agility horses. They are able to travel at faster speeds and carry minimal weight making them good for messengers and merchants
Warmblood Horses would be the average of the two. You have a strong horse that retains a good amount of agility making them excellent for war.
They could be regionally unique, creating a necessity for horse trade and further deepen the experience of having the right equipment for the task at hand.