Okay, here's my opinion on why cavalry developed. Even before Rome fell the Romans were relying more on horses. Why?
Movement: They move much faster than footmen, can carry stuff, and if they die you still have your soldier at least and maybe a little meat. If you have a big border to police or you need to move a bunch of men to spot X fast to be decisive you want horses. For post Roman states, you have more problems with footmen. You need to maintain the Roman roads or build new ones. Horses reduce that problem. Related: If you're expanding east into areas that are pretty wild, you need horses even more. Or how about if your enemies are horse riders themselves, you need horses to catch up to them.
Tactical advantages. A man on a horse vs. guys on the ground as long as they aren't that well trained or even if they are but aren't used to fighting as a unit has an advantage. Not just the mass and strength of the horse that can knock you down, but it's a lot easier to bring more force to beat if you swing both down (direction) and bracing yourself on the horse (more applicable after stirrups but still useful before). Try it yourself, stab/swing up, or swing down. Which feels more natural?
Strategic advantages: If you have tactical advantages thanks to cavalry, then your enemies do not want to face you in the field. What do they do? They retreat to fortifications! Field battles are unpredictable, sieges while boring and miserable are also easier to control. Why wouldn't you prefer from a strategic point a situation where your worries are food/water and not keeping enough men around vs. a field battle that you can lose in an instant due to luck? Take the surer thing!
Population: Small numbers of well trained cavalry can defeat more loosely trained numbers of infantry. So not only can they respond faster than footmen in most cases, you don't have to field as many. You can armor them heavily (relatively more than foot) and being on a horse increases rate of survival by itself. Basically they are easier to organize and hard to kill off.
I think the big things though, are organizational. If you have the money, population and administration to organize, train and take enough men from the fields to fight, you can keep footmen. I still don't see how you can get elite knights out of that absent lack of horses or terrain that made horses more trouble than they were worth to use. It should be noted that in the early modern period, a lot of knights did fight on foot. But it was never like the classic knights whose mystique was bound up in their duties as rulers as well.