Chariot archers were used... just probably not in that particular period...
In China they used 4 horse Chariots that would have 3-4 men on the chariots with usually 2 as bowmen 1 driver and 1 wielding a polearm....
However that sort of warfare also dimished (interestingly around the same time period as the mid east )as calvary became obviously more all rounded with less weakness
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