Lear is crazy, of course -- everyone calls him so right at the start of the play. But isn't Cordelia a little irrational too? Aren't you supposed to humor your dotty elders, not get into an argument over pleasantries ("Why have my sisters husbands, if they say/They love you all?") I almost think that Cordelia is trying to score points against her bigmouthed sisters, rather than paying attention to where Lear is at.
It occurs to me that this is one of the differences between reading and seeing plays; Lear seems too crazy in this scene, as I read it, what with blowing up at Kent and trashing his daughter to the other nobles. (And yet not too crazy to speak in verse; old habits die hard, I guess.) But when you watch a play you don't have time to say "Hey, wait a minute," or the play will barrel right past you.
This is particularly true, I imagine, with great actors in the parts (I've never seen "Lear"), for one thing I like in great acting is the amount of detail that's in it, that's where the surprise in it comes, from an unusual emphasis of a word, or surprise in posture or movement. You'd need that here, I think, to sell Cordelia, who indeed is dragging the whole kingdom down out of some notion of self-righteousness.
(I realize that I sound a little bit like a Wall Street Journal editorial -- why won't Cordelia wear her flag lapel pin? She even marries a Frenchman, making matters even worse. But, considering what does happen, things might have been better for the Lears all around if she had just greased the wheels a little bit.)
UPDATE: I forgot to mention that Shakespeare follows faultless sitcom form by beginning with the B-story -- with Edmund the bastard. You can begin scenes with the B-story, but people don't like it if you end with them.