Cavalier?
"Hmm, I'd say that being rich and deftly navigating feudal obligations is in many ways distinctly un-knightlike, but I'll give you much of the rest."
That all comes down to definitions and concepts. I'm guessing that your schema for a knight would be a chivalric, arthurian guy. I was using knight in the sense of one holding that title within the feudal system, be it historically or in other tales besides those arthurian. Besides which, remember that in arthurian mythology, Mordred was a knight...
As to the idea of a cavalier class, the core problem comes down to this: Is your cavalier more powerful than other classes if he gets to use his horse and about as powerful if not, or is he comparably balanced to other classes while in his forte and feeble from limited options outside his forte? A good example would be the rogue, who's really good under optimal circumstances but badly hosed in tight quarters or in fights where sneak attack doesn't apply.
I won't say that it is impossible to forge a third path besides making the cavalier 'too good mounted and only acceptable otherwise' and 'pretty decent mounted and lame otherwise' but I do think that it is nontrivial to accomplish something other than one of those alternatives.