The early horses were rather small, and not strong enough for a large man to ride them. That's why you see pairs of them pulling chariots instead of being ridden in ancient pictures.
Women were light enough to ride the horses of the day without unduly burdening them. With bows, they could be cavalry archers.
You do know that the era of the Amazons and Hercules predates the invention of stirrups by 5 to 10 centuries? Mentions of horses in the Iliad point to their use as taxicabs, mostly conveying ground forces to a battle location. (I'll grant the forces of Nestor may have had some unique horsemanship skills not shared with their allies.)
As an aside, Greek men did not wear pants, and would have considered any man that (i.e., the Persians) did to be cross-dressing, and consequently make jokes about his masculinity.