I'm going to have to state here as clearly as possible that the movie version of Cavalry riding down helpless infantry was NOT REALITY.
Throughout history, massed cavalry has been generally inferior to massed infantry. Cavalry is effective against open-order infantry (basically, light troops such as archers) or against the flanks or rear of heavy infantry, but the image of the mighty horseman riding down helpless soldiers is just not the way it was.
In the Middle Ages there was a time when infantry (mostly just militia) was so crappy and cavalry (i.e., knights) so heavily armed that they could sometimes defeat massed infantry, but even then it was rare. With the development of trained infantry even that went away. I challenge anyone to name me a battle where cavalry was used in a frontal assault on trained close-order infantry successfully. I can't think of a single case offhand.
From the Swiss Pike victories vs. knights on horseback to the Hussite war wagons trouncing knights on horseback to Napoleon's cavalry being totally unable to break British squares at Waterloo - Cavalry has been unable to take on heavy infantry throughout history, and more particularly in our timeline.
So please stop asking for us to totally invalidate the historical record and have cavalry be able to beat infantry - at least heavy infantry. It just didn't happen.
Yes, one-on-one a horseman is superior to a footman. How often did that happen? How close can two horses charge together? six feet apart? eight feet apart? How consider the infantry - they literally stand shoulder to shoulder to meet a charge. That's one guy for every 2 feet. This means that your single mighty horseman has to ride down 3-4 guys facing him, with pointy bayonets or spears aimed right at his horse's chest. The only way you could possibly succeed would be if the infantry had low morale, and broke and ran away during your charge. This occasionally happened, but not so often during our time period. Even the Aztecs, with no experience vs. cavalry, were able to stand up to mounted charges.