Yes! Go even further beyond the current race minigames and have us have our horses judged by stats and handling - maybe even an arena with combat challenges where you have to do mounted combat only to clear them!
As others have said before me:
Let us board anything we can ride, for a fee (good to have rupee sinks)
Better or more reliable permanent shops
Customize your horse anywhere
Customize your non-horses for a fee (maybe even only at one specialized later-game stable)
Upgrade or improve your...