But... realism is just pointles... flavour should alvays take first place.
Not mentioning that if we want realism, we should dispense with the mages, vampires and the like.
You need to learn about the concept of Immersion Relevant Realism
Mages are alright, because their powercomes from gods, and mana nodes. It would NOT be alright for Warriors to be able to throw nuclear missiles, because that makes no sense. But the argument you're making here is "a, therefore b". You just can't do that.
A chariot is pulled by multiple horses, which are quite likely to get their legs tangled up, and interfere with each other. It's also essentially the earliest form of a car, in a time where spring suspension was not yet invented. Resulting in a very bumpy ride and making chariot archery extremely difficult.
Moreover, a chariot is larger, and less manouevrable than a man on horseback. And also slower. Ever seen the back end of a bus swaying when it turns a corner? chariots are like that. Turn too tightly, it spins out of control and tips over, probably killing the riders. A single man on horseback can turn much tighter arcs, and run rings around a chariot.
Horses are faster, because a chariot is being pulled by them. In the case of a chariot, it's two horses pulling a heavy chariot with two men on it. with a horseman, it's one man, on a horse. Guess which case puts less strain on the horse? The speed of a chariot is also naturally limited by it's contstruction. go too fast and the thing will flall apart from friction. Hell, even just hit a jutting out stone in the ground and very bad things could occur.
Being larger too, comes with a lot of problems. Try manouvreing a chariot through a forest, or a marsh. A man on horseback can do it much more capably and at a faster pace.
moreover, with the invention of stirrups, a horse rider could hold on with his legs, freeing his hands up to swing a sword, or fire arrows. When you're in a rickety chariot, you need to keep one hand on the reins at all times, lest you fall out.
Chariots were used to great effect by the egyptians, certainly. And that was
three thousand years ago., More recently, armoured knights on horseback were in use all over europe in the last millenia. Why do you think they didn't use chariots?