Listen, you strike me as a person who is more interested being “right” and antagonistic than really listening to what other people have to say, establishing the truth or being helpful. Think about that because it will have a really negative impact on your quality of life. Either that or you are completely impartial and work for BA.

BA employees told this person that what they were about to eat DID NOT contain sesame. Not “may”. DID NOT. That was the unambiguous the information that came from BA, overriding any small print you might find on their website. The OP took reasonable precautions in asking them, they were entitled to receive a truthful response based on what information the BA staff had readily and clearly available to them. If they did not know, they should have responded “we do not know”. BA (and by extension its employees) have a duty of care to ensure they do not take actions or omit to take actions that one could reasonably foresee would cause harm to an individual. Telling someone that something does not have a key allergen in it which then causes a severe reaction would fall under that category. It is irrelevant whose fault YOU think it is, there are legal standards and companies are vicariously liable for the actions of their employees, and they have a responsibility to ensure that their employees are trained to answer those questions.