Let's just say that you are a very wealthy family with assets in the country your child was travelling to. A criminal gang who has been monitoring your family and assets decides to take advantage of the travel chaos by raiding those assets when the family is expected to be there, but is delayed. Ringing the airline with some sob story (and bear in mind that the ticketing details etc that you gave are easly hackable by identity theft experts), the airline confirms that all is fine, they will be on a flight tomorrow, so the criminals can take advantage of the delay, safe in the knowledge that they won't be interrupted. Would you be happy that the airline have offered this customer service.
I have personal experience of being identity hi-jacked at a "secure" bank with "secure" systems, and the theft was possible only because a bank staff member responded with non SOP customer service to the criminals who had a sob story.
The fact that BA was so steadfast in refusing you any information encourages me to fly with them.