HA/HX should be part of the mod.
Reducing HX reload speed was important, mainly so that the appeal of the class isn't to be OP, but to fulfill an actual battlefield role.
What I think would be a bit more reasonable is to make the Horse Archery skill never actually give the same accuracy as a dedicated bowmen.
Even if the bow options on horse are not as great as being a dedicated archer, they still need to be less effective somehow.
Traditionally horse archers were only decent at semi long range in large groups firing volleys. HA was meant for a more short to medium range where dedicated bowmen had an advantage of accuracy (not to mention a horse is a big target!)
I am not a balancing expert by all means, I can only go off my experience, but I believe it has to be a combination of either horse bumps changing and / or accuracy on horse being adjusted.
Ideally what I would like to see is horses taking damage for running people over. That way you only charge an infantrymen if its really necessary instead of using it as an abusive game mechanic.