Almost as telling as anything else about these folks' behavior is the OP's observation that the family was loud and had made a terrible mess of food.

Quite obviously, the mother was used to being reprimanded to at least looked at unapprovingly by members of the public, and was being proactive in addressing the OP the way she did.

I'd like to think that a report to the AC staff might have led to a reprimand or a warning regarding loss of AC privileges.