The Magyars are probably a better evaluation of "pure" horse archery in a European context than the Mongols, who as others have noted practiced a more combined arms approach to warfare. The Magyars, in contrast, were almost universally described by the chronicles of the time (both Latin and Byzantine) as fielding armies exclusively of light horsemen. (Infantry were recorded at Lechfeld, but they took no part in the decisive phase of the battle and were probably there only to credibly threaten a siege of Augsburg).
How well did they do? Well, they were good at what we might call "deep raids" - going from Pannonia to Champagne (via Germany) and back again (via Italy) and plundering all the way is a pretty impressive feat, and they managed it repeatedly. Clearly this would not have been possible had the force not been mounted. The bows of the Magyars were certainly noted by their opponents, yet they failed to adopt them. Why?
Firstly, the success of the Magyars was only partially based on horse archery. Arguably more important was their discipline and unit tactics when measured against their enemies. Unlike the Europeans, who tended to gather an army at a mustering point and then march it around as one host, the Magyars traveled widely dispersed over the countryside and only came together on the eve of a battle through the use of messengers and smoke signals. This made them difficult to catch and also caused the Europeans to regularly misinterpret their numbers. They were excellent scouts, and chose their battlefields well. The Byzantines claimed that the Magyars, unlike many raiding peoples, never stopped a battle to plunder an enemy camp, but kept fighting until the foe was completely destroyed. They were adept at playing mind-games too, as Brenta demonstrates; they abjectly pleaded with Berengar to let them leave in peace, lulling him into a sense of security and arrogance even as they planned to encircle and crush his army. Horse archery was a component of their power - their feigned-retreat-and-shoot tactics would not have worked without the bow - but had this been their only asset, it's unlikely they would have made as much of an impression as they did.
As warriors, however, the Magyars were not obviously superior specimens. Whenever the German cavalry actually came to grips with the Magyars, they tended to win, and a Magyar victory wasn't as simple as "just don't let the Germans catch you." For one thing, the terrain was not always conducive to nomadic tactics; additionally, as the Magyars were principally raiders, they were obliged to return with wagons full of their spoils, and the Germans found that the best time to attack them was when they were returning from a raid rather than riding to it. King Henry adapted to Magyar tactics not by taking up the bow or making his cavalry lighter, but by teaching them discipline, preventing them from chasing fruitlessly after feigned retreats and instructing them to charge together without any one cavalryman riding ahead of another. The Germans also adapted to the Magyars strategically by building a system of defense in depth, consisting both of wood-and-earth fortresses from which counterattacks could be launched (the Ungarnwälle) and the organization of local militias which could respond to raiding parties. It was these militias that eventually caused the gravest defeat the Magyars ever faced: Otto's army succeeded only in dispersing the Magyars at Lechfeld, it was the Bavarian local forces who destroyed them in detail as the scattered Magyar bands attempted to retreat over swollen rivers and fords throughout the country. By the mid-10th century, the Germans were regularly beating the Magyars in the field and were even raiding Magyar territory in retaliation, even though their equipment had hardly changed in the last century.
The Magyar horse archer was good for a certain kind of warfare, the "deep raid" that avoided fortified places and struck at the soft underbelly of distant kingdoms. Ottonian warfare, however, was based around the siege - most battles of this period in central/western Europe took place in the context of a siege or an attempt to relieve a siege. Besieging required the fielding of large armies with many infantrymen, and prioritized logistics and staying power over speed and surprise. The Magyars were frankly rubbish at sieges and generally followed the path of least resistance; despite having practically free reign in northern Italy, they only ever succeeded in sacking one city. The Magyar horse archer, unlike the German miles, was unsuited for the kind of warfare that the Germans and other Latin-Europeans generally engaged in. Even if you ignore the cultural factors - which you probably shouldn't - this is sufficient to explain why horse archers were not adopted by the Germans and their neighbors even though the Magyars did humiliate them on several occasions in the late 9th/early 10th centuries. The horse archer "weapon system" was not strictly better than what the Germans had, and Magyar successes had more to do with tactics and stratagems that could be adapted to and effectively opposed without any shift in equipment.
The Magyars themselves, having been bested, ended up settling down and within two centuries or so were fielding pretty much the same armies as their Latin neighbors - and defended against the Mongols at Mohi with a thoroughly "European" army.
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