Mounted archers are perhaps best known for their use by the various Eurasian nomads who made their homes on the vast Eurasian steppe. Notably, the tactic didn't develop out of some advantage it provided in extended warfare. It most likely developed out of hunting and herd-protection techniques among the various "horse people" cultures, particularly among the Iranian and Turkic peoples. In this sense, it's hardly surprising that agrarian Europe would not have independently developed this tactic or made extensive use of it.

So then, once they had contact with mounted archers, why not make use of this tactic themselves? Well, they did! Mounted crossbowmen were used by Scandinavian and German armies. There are also recorded instances of the French making use of mounted archer mercenaries throughout the medieval period. They were typically used in a skirmishing capacity for reasons detailed below.

However, it is true that this practice never really took off in Western Europe as it had elsewhere. In this regard, I'll admit to some speculation since the question seems to invite some hypothetical posturing. Mounted archery requires that the rider have fairly significant equestrian skills since it involves a good amount of hands-free riding. While this would have been necessary among most members of clans on the Eurasian steppe (which also had a fairly strange class structure), this level of equestrian skill was not at all common among the poorer bulk of Western European armies (hence the use of mercenaries with the skill). Additionally, as these ideas spread throughout Western Europe, they were also quickly developing the massed crossbowmen tactic which is fairly devastating to mounted archers who present a faster moving, but larger and slower-firing target.

I must apologise for the lack of specific sourcing. However, I can point you in the right direction. Most of this was cobbled together from Chris Harman's 'A People's History of the World', particularly regarding technology advance in Europe and Mongol social organisation. David Anthony's 'The Horse, the Wheel, and Language: How Bronze-Age Riders from the Eurasian Steppes Shaped the Modern World' is also useful for details on how extensive equestrian skills developed, their social importance, etc.

Western Europe's terrain is unsuitable for those kind of tactics. Hit-and-run doesn't exactly work well in the forest, hills or mountains.