The issue with horse archery is that it has some major downsides. Foot archers tend to outshoot them and massed horse archers tend to lose against combined troops. European armies also had fairly good success against horse archers as the Romans defeated the Parthian horse archers on many different occasions. Europeans knew about horse archery for a relatively long time but never really chose to adopt in to the extent of other societies.

Horse archers excel at harassing the enemy and engaging in skirmishing tactics. This adoption of skirmishing tactic is also one of the main reasons why you don't see horse archers in most of Europe. Skirmishing requires fairly open areas to allow the harassing troops to retreat. Large parts of Europe are heavily forested / hilly and are not open plains meaning that horse archers could more easily be charged and defeated.

That being said, some medieval European armies did utilize mounted crossbowmen. Later European armies would also use mounted gunmen in combat which is also similar in ideology to horse archery.

Geography mostly.

Horse archers were usually tribel people that lived on the steppe, now if you look at the great steppe you will see that it stoppes right befor europe and then there are the Carpathian Mountains making a nice shild. Unfortunatly for europe there is a little steppe ofshoot behind these mountains, now called the hungarian steppe, and that is where all the great steppe empires set up shop and raided the west. While these raids were possible, they never really had the same power as they did on the steppes. Northern europe was full of woods, then you have the alpes and so on. So the utility of a horse archer goes down somewhat.

In europe horses were always a luxury good. You can even see that in the organisation of the early roman army. You can see it in Alexanders army. The same goes for the middle ages. This alone would not make the devlopment impossible, even a small mount of horse archers are better then non at all. Another thing is that horse archery is extremly hard and training men to do it, is costly. The only major empires who actually trained natives as horse archers, are the Byzantians (around Justinians time) and they only had small numbers and prefered using huns or avars.

There are more problems. The weapon that really changed the amount of danger coming from horse archers is the power of the bow. The romans allready had trouple with the scythians but the huns were on a hole other class, they seamed to use a better bow. These kinds of bows were hard to make and so far as I know the secret of making composite bows of that style never really arrived in the empires around the steppe. They used bows like it but did not make them. This is another advantage of the tribes living on the steppe, they needed these bows and lots of people could make them.

One last point on tribal organisation. They way these tribes thougth was very dynamic, more like a swarm and that did help them in battle. They could break up, change there direction and all kind of movments that really required a amount of coordination that western nobels could not match. Even western nobels who trained fighing on horses from a jung age, had huge problems to pull of a charge. The sort of dynamic hit and run with ambushes was really not something you can learn, if you are not living on horses together as a army or a tribe.

tl:dr; Not enougth horses, lacking the bow technology, to costly to train, to hard to organise