A similar problem exists with elephants - the crew dying only when the elephant is dead. This is fine for the more basic form of elephants, but is a little unfair for the more elite elephant units (which also happen to include arrows).

If it were possible, it would have been nice for chariots (and elephants) to act like two+ independent units which, to start with, are fixed onto each other. These units would, obviously, be the rider(s), the beast(s) and (in the case of chariots) the cart.

If the beast(s) are killed, the charioteers/elephant riders, could jump off and begin fighting in melee. Likewise, if the riders are killed, the beasts, as said, could just run around slaughtering anyway. Maybe enemy units could take control of the surviving unit, calm the beasts and use them to their own advantage if they so wish. If the cart was destroyed (with the charioteers on board), which could realistically only happen in melee, the riders would die, and the horses probably would be trapped by ropes restraining them.

Before anybody even is killed, however, it would have been nice it it was possible for the charioteers to simply jump of their chariots (leaving one man on each one for security reasons of course), and attack nearby units. Then reboard and go wherever else they are needed. I believe the ancient British boarded their elites onto chariots like this in their battles.