Did they delay the flight while the child was supposely sick? If so, that I don't understand. Now if the psycho's child was considered by the paramedics ok to travel before boarding commenced, then I can understand the Captain checking operations for advise or consent. The last thing AA would need is a media storm of "flight takes off without woman and her two children."
But I agree she sounds like a real pain in the a...s and I am glad that she got booed when she boarded.