"Court" I think, in this sense, was the king's court, not a judicial court. But, I think you're onto something about how it started as someone who took care of horses then probably evolved to being in charge of horses or something, to then being the official with the big job related to horses. I guess it's a bit like how "secretary" can be either a person making near minimum wage who does office work, or it can be the most powerful person in the cabinet who deals with X.