I think your pros/cons of cavalry could use a rewrite:
The good:
1. They can pick their engagements with high mobility.
2. Works well against lone and scattered or fleeing enemies.
3. A great tool to create distractions.
4. Heavy cavalry can bust up infantry clusters if they forgot to bring spears.
The bad:
1. Unarmored horses unable to attack groups of people.
2. Dead when dismounted, unless it was by another cav or archer.
3. Charging knocks people over, but does little damage unless target unarmored, even with war horses.
4. It is expensive.
5. Map does not always permit effective use of cavalry, meaning you have to go on foot and be gimped for having +5/6 Riding or whatever.
6. Horses are easily killed.
7. Pikes are probably the easiest weapons to use in the game, and the most deadly in skilled hands.
Cav isn't OP, cav is a force multiplier, the reason they get so many kills is because most battles are very spread out and disorganized, the Cav is basically a vulture/kill-stealing class. Basically the Rogue in any other sort of game.