Now I know a lot of you are going to say it is OP and in some ways it is but there are a lot of weaknesses to horse archers which I will list below.
1. Countered by foot archers.
Foot archers take up much less space than horse archers thus can form much tighter formations and fire but more concentrated volleys. Also horse archers cannot use large bows so generally they will have a shorter range. A horse archer is also a much larger target due to the horse.
2. Cannot fight on difficult terrain.
A horse archer cannot fight effectively in woods because of trees which run the risk of riding into and also they block vision so the archery is ineffective. They are also much larger so cannot fit into tight spaces between the trees like foot archers can. A horse archer will also run into difficulty in mountainous terrain as cavalry will struggle to move up hills and such without slowing down and getting tired.
3. Not effective in sieges.
A horse archer is a larger target, and has a shorter range than normal archers. Also, you cannot kite and stationary object like a castle. Foot troops are required here.
4. Difficult to learn skill.
You need expert horse handling and archery skills to be able to perform this tactic. Which takes times.
Horse archers would not break the game. Like all other military tactics, you have advantages and disadvantages, you just need to play your cards correctly against them.
A potential way the game could implement this is similar to the way the game does shields and spears. Your character would need 90 in cavalry to unlock mounted archery, and to use the bow you'd need archer skill as normal. It'd already be immensely difficult because archery would have to be done in first person while controlling the horse. This would mean you would essentially have to control WASD while aiming the mouse and shooting at the right time, all the while making sure you are not going to run into anything. If you are going fast you'd also have to deal with camera shaking. It's already honestly hard enough riding without hitting any trees in this game, let alone in a unit with several other horseman you have to take care not to hit/block off.
I feel horse archery would improve many players experience as the Khoor race is based on the Mongols and would benefit greatly from this addition.