Exactly how cavalry was used in the high Middle Ages, how useful it was, and how its role changed in the later Middle Ages and the early modern period is a difficult question, or series of questions. However, I feel on very solid ground to assert that horsemanship and fighting on horse back were considered by contemporaries to be an essential element of noble identity. A man at arms, a warrior of high standing, was a cavalryman, and – here's a practical note – was paid more than men on foot, gens de pie, were.
Here's another interesting story relevant to this matter. It comes of course from my favorite text, the Chronicle of the Good Duke, written in 1429 but in this case claiming to report an incident of 1388. Once again the Duke of Brittany and the Constable of France were fighting. The Duke had brought up a substantial army and the Constable was outnumbered. Here's what happened next, according (probably) to a survivor of this campaign, Jean de Chastelmorand.
The Duke of Brittany seeing the battle order of the Lord of Clisson told his men, "My lords and companions, see Clisson there, who has arranged his companies and desires nothing but battle. I would not refuse it at all. willingly but I see that he has put together a great wing of his men who are mounted on great coursers of superior quality. Our horses are small; those over there will come charge us and we will not be able to withstand them; and things will be the worse for us."Some of my readers will know the catchphrase "who gets the horse?" that came out of recent discussions of Charny's questions. Here we have "who has the heavy horses?"
Update: somewhat later, the Lord of Clisson says something of this sort:
Beaumanoir will lead the remaining men of my household and I don't want you to have more than about 150 men at arms for you are plenty for 600 horses of Bretons. And I swear to God that the horse of the Bretons are worth nothing and it seems to me that you will not fail to find your adventure and you will be able to take it looking good and complete it.Further update:
When it came to the middle of the day, the Lord of Clisson said to his men, “My good Lords, you are well mounted, the horses of those over there are small; charge into these Bretons and push them into the battalion of the Duke.” And they did it just so and in this melee both a good 100 Bretons of the Duke’s party were killed and a good 100 horses gained;