Simply put, there are several reasons why you'd want mounted infantry over just lots of cavalry. The first was the nature of warfare in this period tended to emphasise raiding (or chevauncheé) where the aim was to damage your opponents lands and enrich yourself and your soldiers. This inclination had existed previously but became an essential part of warfare as Europe become increasingly dense with castles which could not be taken easily with long sieges and would be a threat to slow moving forces deep within enemy territory. Mounted soldiers were also able to range and scout or pillage far further from the main body and thus provided essential information and victuals for a raiding force, as well as allowing them to regroup quickly in battle.

Pitched battle as you describe was the last resort of many campaigns as many factors which affected victory were simply outside of a leader's control. It was a gamble, occasionally a calculated one, but still a gamble. Moreover, when on chevauncheé an army which was caught or elected to do battle would often be vastly outnumbered. Better to keep moving unless you can take up an entrenched position (where you'd dismount most of your soldiers anyway - the English tactics at Crécy, Poitiers, and Agincourt in the fourteenth- and fifteenth-centuries). The army would be travelling with wagons or supply mounts anyway, so the logistics to protect these were already in place, just slightly increased.

The importance​ of balanced forces has repeatedly been stressed by military historians such as Matthew Bennett (I forget the exact citation but the article is called 'The Myth of the Sumpremacy of Knightly Cavalry', I think Harlaxton, 1994). Armies which were imbalanced or didn't utilise a good combination of infantry, cavalry, and missile units tended to lose battles they fought, so there was good reason to keep mobile infantry and archers in your force.

Another reason was that war horses were far more expensive than the rounceys which were more typically ridden and keeping your riding horses out of battle meant that they were less likely to die and thus hamstring your mobility.

Finally, training soldiers to act as cavalry was extremely time-consuming and expensive. The charge was a highly coordinated manoeuvre which required immense discipline and riding skill (alongside appropriate mounts and arms). The common soldiery simply were not equipped or prepared to make that transition until the late thirteenth- or fourteenth-centuries.

If you're interested in more, it would be worth reading John France, Western Warfare in the Age of the Crusades, 1000-1300, (Ithaca, 1999) or, if you prefer to source something from the internet, De re militari has numerous of varying articles on medieval military history, although be warned that these can be confusing without pertinent historiographical context.

Written on mobile, please excuse formatting or autocorrect errors.