Truth of the above or not, and it's mostly true,
Historically, it's safe to assume people used cavalry because it worked, a lot. it's terrifying. All the training in the world won't keep most people from turning to bolt when a row of horses are coming AT them. Most of an army is not knights or professional soldiers, it's your average Josephus, Hans or Mehmet with a sharpened stick, and he doesn't wanna DIE. The instinct to preserve their lives is too strong.
If you think it wouldn't work on you, go stand next to a real horse trotting past. Not a Shetland pony at the County Fair that you "rode" when you were eight, I mean something 12 hands or up. (Side point- Medieval horses, even the destriers were quite small by modern ideas. A destrier is probably the size of, and built like, a 12-maaayyyybe14 hand AQH or sporthorse. A bit blocky but sturdy and able to spin on a dime.)
Anyway, go experience a HUGE intelligent mammal carrying maybe the one person it loves above all else while barreling past you, not even at you, and you'll see why having a horse charge at you with the intent to trample over your broken corpse is a pantswetting experience. (Even if you know horses rarely step on people on purpose, you know you will be going down.)
If you add archery to this horrifying advance, skilled archers able to kill with precision in the advance AND the retreat FROM a moving horse, and you'll see how Chingis Khan did what he did.
I got knocked down by a horse once, and it ran overtop of me. I was certain the whole herd was coming, and that I was going to die... and I knew that all I could do was lie there.
Thankfully it was only the one horse, who had been spooked by another horse. But even though I was only 10 or so, I instantly went home, knowing I had nearly gotten myself killed. I will never forget that... the certainty that I was about to die and that I couldn't do anything to stop it
