So having a horse is cool, but of course what everyone wants is black horses. Except me. I want a white one. To those who know me, this will come as no surprise.
White and black horses will come, surely, but they're clearly being saved for later, probably as another cool thing which high level people can use to set them apart. Perhaps they will even be as fast or faster than the express pony system.
Now, if I were a developer, this is what I would do for black horses. We actually know, from the Riders of Rohan, that orcs raid the horse lords and they always steal the black horses.
So you can see the quest here, right? Maybe you have to protect a black horse from orcs. Or you need to go into an orc camp, rescue a black horse that has been stolen, and lead it back.
While we're on the subject, we also know that in order to ride Shadowfax, Gandalf had to follow the horse across Rohan for a few days. How he kept up with the stallion is beyond my ken, but it's best not to get too nosy when you're dealing with a grouchy wizard. In any case, following the horse around until you wear it out enough to approach also sounds like an interesting quest, if less combat intensive than the Black Horse Rescue.
That's what I would do. If I were a dev.