So far as you are aware of the limitation, everything is peachy.

With crossbows, the main advantage is simplicity. Most cultures that employed mounted archers had them train from the time they were children. Even the best bowman on foot, like the longbowman of England and Wales, required years of training to be effective. A crossbow, on the other hand, can be used by any dumb motherfucker who can look at something and pull a trigger. I know this, because I own a crossbow. My father also happens to own a longbow. Both are modern fiberglass builds, but even then at least the differences between the two styles are noticeable. 

If the skilled professional warriors are all doing something else in his culture, then it is possible mounted archers are being recruited from a lower class of riders. If they are not proficient with a bow, they won't be shooting bows from horseback and a crossbow (which doesn't require proficiency) could easily be substituted.