It always amuses me that gate agents often use the "everyone else" heard the announcement as if they are suggesting that the person was lying. If they heard the announcement, do you think they deliberately missed the flight; decided to ignore it or had cooked up some dastardly scheme to make Delta look bad?
It is irrelevant whether anyone else heard it.. what is relevant is that they relied on the advice of the airline of the revised departure time. If they had indeed been told it might be brought forward and to remain either "at the gate" or "in the terminal", fair enough. But often they are not.. and indeed I have often been in a situation in which the delay is posted on the screen, with no further information.
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