First off, is the question even right in the first place? Is the emphasis on mounted knights just a product of propaganda or does it have weight? Second, did cavalry actually achieve significant enough advancement in the medieval era to advantage itself over infantry? I am aware of challenges to the notion that stirrups were what enabled mounted cavalry in warfare. Finally, was the knowledge of doctrine from the Greeks and Romans 'lost'? I find it hard to believe that there was a complete and total scholarly void that didn't look to the age of antiquity.
In the reading I've done, it suggests that the use of cavalry was a product from adapting to the feudal system. Given a relative lack of centralized control, coordination and wealth, the large professional and semi-professional armies of Rome and Greece were impossible in the smaller and resource-light fiefdoms of Western Europe, since lords couldn't pull away a large majority of their labor for extended periods of the time to effectively train a well-disciplined force of heavy infantry, but could afford to upkeep a smaller corps of lifetime fighters that could also patrol their lands.
Western Europe is heavily forested during the Medieval period, without the ready availability of horses enjoyed by, say, Steppe peoples. As such, maintaining a cavalry force requires stabling and infrastructure and horse breeders, and so for much of the early Medieval period, small kingdoms rely largely on infantry levies while heavy cavalry remains the prerogative of larger states like the Carolingian Empire. Heavy shock cavalry are an incredibly effective weapon of war though, and as such were heavily desirable. Otto I, according to Regino of Prüm's Chronicle, paid off raiding Magyars for many years until he was able to construct a series of fortifications throughout Germany, consolidate political control and properly train and equip his mounted loricati. They showed their value at the battles of Lechfeld and Recknitz when they utterly routed the light cavalry forces of the Magyars and the Wends who had traditionally been able to outmanouevre infantry forces.
Cavalry are also able to control a wider area than infantry by virtue of their greater mobility, as you suggested, allowing a smaller cavalry force to patrol an area that would have previously required a more substantial infantry garrison.
(Bartlett, The Making of Europe; Charles-Edwards, After Rome.)