It seems to me that having the speed of a horse and not having to tire your legs by running and marching in battle while being on a raised and wheeled platform as opposed to the horses back, with a division of labour between the man doing the fighting and the man driving the horse, would be in many ways advantageous over the mounted cavalryman, but it seems that even by the time of the Hellenistic Age and certainly by the late Roman Republic the war chariot was considered out of date and was reduced to the equivalent of a Formula 1 vehicle, purely for sporting. Why?
There's many other difficulties you've overlooked that make the chariot a more fraught choice than just riding the horse. Most notable is the well-known fact that in the pairing of horse and man to create a cavalryman, the horse has all the brains. Plus two (or more) horses is always going to be more expensive than one horse plus rider. More can always be said about chariots and their intricacies, so if anyone else would like to address themselves to the topic, please don't let this post stop you!
For the meantime, OP, here are some previous posts on chariots for your perusal: