An alternative to, not a replacement for, the rope tether for horse, which which takes up two inventory slots and is constantly grabbed first by the game whenever you build anything requiring rope.
Reins would be made from leather straps (maybe also a metal nugget) and would, like saddlebags, become part of the horse. When you dismount the horse is tethered to you by the reins, when you click on a hitching post the horse is tethered there by the reins. And like saddlebags, reins would be removable. So once you can make reins there is no need to keep a rope in inventory.