Remaking cavalry should be focused around one principle - cavalry charge is the main weapon, not zig-zagging between enemies and stabbing unaware people. More like actual horses, less helicopter gunship.
Therefore, riders should have a triggerable "charge" ability with long cooldown, which accelerates the horse, makes it a lot less manueverable but also the bump should be deadly and the horse itself way less vulnerable during it to anything except anti-cavalry weapons. So group of riders charging together means there's teeth and bones to collect if they catch spear-less infantry in open fields. Something similar to current "shield +1" etc. mechanics could be used so that horses which are charging in group don't instantly die from fatal wounds, though they do die after completing the charge. On the other hand, if cavalryman makes wrong decision and commits to a charge against pikes, he will die since he won't be able to insta-stop and ride away.
This obviously requires 1) making lone horseman way less maneuverable and durable when not charging than he is now 2) removing attacking while in air for infantry and general reduction of jump height and distance. Less monkeying, more thinking, on both sides.