I think it is a good idea to have your mount have a mind of their own. My riding teacher used to tell me that riding a horse requires both the horse's training and the rider's skill. If either lacks in one, the ride will be lessened, possibly even disastrous.
A poorly trained horse with an inept rider can be an accident waiting to happen. However if you have a master rider, with a poorly trained horse, the horse will still do their own thing, but the rider can normally get the horse to follow basic commands, with an eventuality of the horse being slightly better trained by the time the ride is over.
But a horse that has been trained by a master, with an inept rider, will still follow commands, but they will end up spending most of their time trying to figure out what commands you are giving them, and also keeping you on their back. Most people probably don't know this, but a well trained horse will do their best to keep you from falling off. Going so far as to completely stop and side step or lean, to help you regain your balance. With worst case scenarios being that the horse knows you will fall, and will either get away from objects that might hurt you in the fall, or bring you to a tall object so you can grab on to it. Both I have experienced numerous times when starting out.
My trainer always had a philosophy of training new riders on old and well trained horses, and then as the rider's skill increased, he would swap out the old horse for a slightly younger one, then to a young one with slightly less training, and so on all the way down to eventually, letting his student's ride untrained horses if he believed them skilled enough.
Something to keep in mind, horses do have a mind of their own, and their own interests, which may not always coincide with the rider. The only difference between a poorly trained horse and a master trained horse in this matter, is the fact that a poorly trained horse will act on their urges, while a master trained horse will give some indication on what it wants to do, and let you decide some there.
For instance, the horse I used to ride was a pinto mare named Happy (she was named by my riding teacher's granddaughter). Happy was trained to signal her intentions with her head. Which required me to carry a notepad at first with all of the different meanings. Head up and held, meant she wanted to slow down. Nose pointed straight and held, meant she wanted to go faster. Head slightly down and held, meant she wanted to stop (usually to eat or drink). You get the idea. While Happy was specially trained to communicate her intentions in this manner, most horses don't, even with the same level of training. But they do have mannerisms you can pick up on that will tell you what they want.
Also, I wanted to add that the age of the horse can have major effects on their personalities. Happy for instance, was in her 20s when I rode her, so she liked going slow and steady, avoiding a gallop if she could (requiring me to indicate that I wanted a gallop a couple times before she would go into one). Where as when she was much younger, my trainer told me that she was perhaps the most active and energetic of all his horses, constantly wanting to jump straight into a gallop from a trot. In her youth, she hated going slow, and loved to mess with her rider by turning unexpectedly. Or on the occasion that the rider would keep her from galloping, she would put her head down suddenly, stopping in the process, and wouldn't move until the rider got off to check on her, with her bursting into a gallop the second the rider was off.
Now, I doubt most of this can be programmed easily, but I think it would awesome to see at least some of it in the game. If you add personalities and needs to a horse, it brings more to being a master rider than being able to ride better tier horses. It also brings more to training horses than just some numbers, requiring trainers to find and break bad habits, or engender good ones.
