One of the coolest thing I did in Canada when I lived there was learning how to ride horses. I worked in a farm with a family and the mother was a professional trainer that worked with badly treated horses (people use to be assholes with horses, they use to jab the spurs till the animal bleeds, hit them and even run over them with cars), horses by nature are shy and sensitive so those animals were even more distrustfull.
Im telling you this cause what you need to tame an train a horse in order to ride them is not only a leash, a saddle and some horseshoes, its a process that involves not only the psicological part between the man and the beast (a relationship of trust need to be built) but a lot of time of riding inside fences for the horse to understand the man and viceversa.
So at least, if the devs want to add the feature of riding horses, I want to see some fences and maybe a stable and some form of tempting the wild horse to get into them, leaving a trail of apples, carrots or something. An item that, after being dropped in the ground, attracts the horse if the players are far enough. Then, with the horse inside the fences, after some time of feeding it and leting him alone then the horse lets you put a leash, some horseshoes, a saddle or whatever you are able to craft and equip on them so you can ride.
I think it is important for the game, if the devs implement a feature about riding horses, to exist a time and effort difference between a wild horse and a trained horse, so trained horses are something valuable to steal or to kill (if the devs are cruel enough to implement anything of what I am saying I am gonna cry so hard if someone kills my trained horses). The horse was a weapon IRL, it can be one easily in Rust, the horse offers a buff of speed and carrying space over the regular player. Even if cars get implemented the horse is a cheap alternative.
About upgrading, I dont think it works with Rust, the game has an equipment system so it would be intuitive for the horse to have his own inventory so you can equip the best gear you have available for the horse and that gear came from crafting or raiding.