I think it's noteworthy that while in game I can beat archer cav units even solo (Once they're out of arrows, every unit is just a pike stab away from getting killed). If you auto-calc horse archers will take out you're entire army, even if there's just 4 units, they can eliminate you. I lost my entire army without taking out a single unit and I have to fight them in battle if I want to win.
I know typically autocalc isn't a fair assessment, but it seems it could use some balancing.
While I do feel like horse archer cav is accurate in game, it isn't broken. What I do feel is that these units need to be expensive. Countering with infantry is a pain because you're usually guaranteed losses. Countering with archers isn't always feasible because it's moving target vs static target. Countering with light cav sucks because light cav aren't tanky enough, and heavy cav is slow.
A few suggestions on balancing then.
Rather than slowing horse archers, I would recommend horse bows have slower missle speed, and long bows have faster, so that standing archers are more capable of hitting horse archers.
Light cav could use some manueverability bonus's in order to reach horse archers with clever positioning. Perhaps just a bit more in horseman skill training could work.
Infantry A.I. really needs work on how to fight horses. They seem to be pretty good against the player, but don't seem to thrust when charged against by A.I. Maybe it's just luck, I dunno.
Formations can help, eg. Shieldwall is good, but when charged, it's thin so horse just runs through it. Circle opens up the back side of units to arrows, but good in concept. I think a Phalanx formation or would be really useful (essentially a tighter square formation that prioritises polearms and shields). Alternatively, a testudo formation would be similarly good, and allow the infantry to rotate to stay safe, I've tried using skein formation with mixed results.
Horse archers aren't invincible, we just need some balancing to happen so they're not so amazingly cost effective.