Gameplay wise this gives us two-four mechanics for people ontop of mounts.
1) Natural Aim difficulty for attacking while on mounts
2) Bodyplacement for mounting, hands required
3) Movement alterations for body placement
4) Chance movements (moving the mount in specific ways to improve aiming but little else)
maybe more
1)If I'm moving with mount, it doesn't mean automatically that it's harder to aim than when standing still. If I'm in a car and have to shoot someone from inside the car, it'll be fairly easy if I can stand trough the roofhole & car is going towards the person. It may actually be easier since there's some support for my hands to lie on. Besides even if it's bit harder with training that "on mount" -handicap could be negated completely. Besides not all mounts are like horses - jumpy and unsteady.
2)As in, you can't use hands while mounted? Nah you don't really need that. Like I said earlier, not all mounts are unsteady and jumpy. And you can very well hold yourself on horse with legs alons. No "hands needed" requirement for mounting, please.
The points of 3-4 are bit odd, but overall, the idea of "mounting handicaps" does not really sound good to me.
A legendary mounter might not have handicaps, but a dabbling mounter certainly would. Not everyone can mount successfully on their first try, let alone do so hands-free!
But enough innuendo... more to the point, firing while mounted is much more difficult than stationary, and I challenge you to find a counter-example - some animal that improves its riders' performance in archery competitions, compared to being on foot. Even modern cars with shock-absorbing suspensions and wheeled locomotion (as opposed to n-pedal) on modern smooth roads, when going straight rather than maneuvering as in combat, are still going to mess up your aim! The vibrations, airspeed, parallax, and just generally more complicated dynamic projectile ballistics all conspire to make it really hard. Add in moving evasively on a rough battlefield and using a relatively slow projectile, and it becomes a nightmare even if the mount is docile - which it won't be, after adorning its flanks with a couple hostile shafts.
Shooting while moving is inherently more difficult even under ideal circumstances with a perfect mount. No mount is perfect, and combat circumstances are rarely ideal.