I think it's best to state just how challenging mounted archery actually is. The people of the Steppes began, in some societies (IE, Mongols), riding horses at a very young age, possibly as young as three or four to begin training. The geographical location makes it more logical as to why; the Steppes are huge, and very vastly flat, making movements on horse very easy, and also very important to the survival of these cultures, since most were nomadic. As these societies developed animal husbandry and then archery, and put the two together, they definitely would have their children practice mounted archery at a very young age. This is also not to say there is no evidence of early European societies not using mounted archery. In fact, there is written evidence that Macedonia used horse archers at some point. It was remarked that at the battle of the Jaxartes, Alexander used mounted archers, though it is probable most of them were local natives forced into conscription. By the time of the infamous Mongol invasion, Western Europe was well into the Middle Ages, and a lot of societies adopted feudalistic societies. It is important to note that feudalism wasn't just a way of governing and fighting; it was, also, a lifestyle. It is perplexing to see why, after such disastrous defeat at Agincourt, the French would not abandon such chivalrous knight tactics that they used at Crecy? Feudalism and chivalry was a lifestyle, not in all of Europe, but in the majority of the Mainland, it was hard for people to give up on a specific way of living, as it would be for any populace. Individual perceptions of glory and honor can go a long way for a lot of people, especially when that's the mindset they are raised in. As for early roots, Rome never used cavalry as extensively as they relied on their legionary foot soldiers. Peter Heather explains in his book, "The Restoration of Rome: Imperial Pretenders and Barbarian Popes" that the origin of extensive use of heavy cavalry within Western Europe, possibly began under the fighting systems organized by Charles Martel, a famous Frank. Over time, the model of heavy cavalry used for charging infantry was an engraved military mindset in the area, and since the Franks would be the ones to form the Kingdom of Germany (East Francia), this would transfer over to the latter Holy Roman Empire. In Spain however, we have a much different scenario due to the great amount of Islamic influence from the Umayyad conquest of Iberia. Here we have examples of Western Europeans using mounted Archery as far as the 15th century. These units were called 'Jinete' which is Spanish simply for horseman. According to Sir Charles Oman, the Spanish used Jinetes to harass the enemy lines, and then charge when a chance was given.

They had institutions that allowed them to do that. There was a very efficient centralized taxation system, there was conscription, and the training of the conscripts was extensive. There was also a very high literacy rate for the era in the Empire (certainly way higher than Western Europe) which made the training even more efficient with the use of military guidebooks.

Becoming a mounted archer was the most difficult task and it required a lot of training and high skill. The soldier firstly had to learn how to use the composite bow. The composite bow was a very demanding weapon and to learn how to use it while riding a horse was extremely difficult -the Huns(who introduced it and shown its power) were able to do this because they were growing up with it! The mounted archers were being thoroughly trained for years, and there were detailed war manuals describing this.

So, the answer is that the West was lacking the infrastructure and the institutions to produce mounted archer units.

(The Grand Strategy of the Byzantine Empire by Luttwak contains most of the above)

This has to do with geographic location. A horse mounted archer is extremely effective on open plains. Originally designed to heard animals, an archer on a horse had a great sight line, great mobility and a better range than anyone on foot. The Eurasian steppes were a perfect place to birth this type of culture.

Europe, on the other hand, is/was heavily forested in the north and fairly rugged in the south. Sight lines are poor, the terrain is dangerous at a full gallop and it's far more agrarian than nomadic. Tight terrain allows light cavalry to be charged easily and can be defeated with infantry and crossbows.

Horse archers could also be countered with mass foot archers. An archer on foot could carry a bigger bow with a longer range.

It's interesting to note that many German armies would have a mounted crossbow unit for scouting, harassment and for a point blank shot charge.