I can see on pg 115 you can still ride a mount in a non-open zone you just loose the bonus movement, but why would you not be able to use a bow when you have horse back fighting. Pg 76 it says ot only works if you are in an open zone. I could get no charge but why not shoot while standing there.
The idea is that you move all the time if you are on a horse, to take the advantage to be on a horse (like circle the wagon). It is more complicated to use a bow from horseback than to fight in melee from horseback so I guess this is to reflect that. If you want to allow this, do it. It will make horse archers more dangerous.
Also if you had fast shooter 3 and horse fighting 1 could you use the mount movement of 2 to run while you attacked as 2 fast actions. And when would the attack take place. Since you can move and shoot what range does the shot take place at. You could start at long (normally-3 ranged) run up to near (+0) shoot as the same fast. Then shoot again at near and run back to long. Without the mount you could still do that but only back to short (-1). Not as tricky as the 4 attacks with a sword but still very good
This is the big bonus being on a horse in combat. So, yes, if you at Long range, you could do this:
* Slow or Fast action. Use Fast Shooter 2 to move and shoot. The horse has a movement of 2 so you will move to Near. You use Fast Shooter 1 so you can shoot without preparing the bow.
* Fast action. Use Fast Shooter 2 to attack and move. You use Fast Shooter 1 so you can shoot without preparing the bow. You then move back to Long range agaub.
... and now we know why the Mongolians where so effective.