CBP state clearly on their site that the ESTA is valid for entry up to the expiry date, even that it can expire whilst you are in the USA without problems.

AA check-in recently told me I couldn't fly if there was less than 6 months left on it, though they didn't ask to see it to check.

Is there really a special AA rule, or did the agent just misunderstand the question he was being directed to ask me?