Well again, we could have just said "eh, just roleplay it!" to a *lot* of current (and good) game mechanics, but this seems to me a game that enjoys creative and fun little new features.
With horses your travel time is decreased. If all your infantry rides horses the entire unit might travel at least twice as many miles per day. If you're a noble you can choose to ride a horse or not, but if you don't you travel slower. All of which is best handled by some simple mechanics rather than pure roleplay, IMO.
I think it could easily add something to the game. It's part flavor, part gameplay change. If you and your infantry really do all have horses (in-game), or you're all cavalry, you ought to get significant movement bonuses compared to foot soldiers. You could still RP the details. Maybe you have a spotted brown palfrey.
(As for ox-drawn carts, the drawback with going there is .. they're
cows. Whereas nearly every depiction of medieval (or a variety of timespans really) nobility has them riding horses. And this makes sense because of the warrior based society following the Roman Empire's collapse, which turns out to be so much like other warrior societies for example the Mongols: No one cares what oxen you have, but being a horse owner and rider is a status signifier.
So I don't think having individual oxen is worth it. Though it is interesting to think of carts as being (specifically) ox-drawn. They would make at best something like 10 miles a day. You could slaughter the cow and get meat... that might be something to consider for the new economy and resource system.
)