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.
You do know that the era of the Amazons and Hercules predates the invention of stirrups by 5 to 10 centuries? And yet the early Greeks and Romans used cavalry before the stirrup came to Europe. Yes, riding a horse without stirrups and modern saddles makes it harder to stay on and ride well, but that doesn't mean that you can't ride.