They are simply refusing to reply to the fundamental issue. When you swiped your card at the check in machine, you began the check in process. YOU WERE THEREFORE NOT LATE FOR CHECK IN. As such, i think you are entitled to denied boarding compensation for an over-booked flight.
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it is not always possible to forecast the number of customers who will need assistance
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What are they talking about? Their computers list everyone who is booked on each flight, and they know how many flights they have booked. Of all industries who may face unusual customer demand, airlines are by far the best equipped with data telling them precisely how many people will demand their servcies and staff accordingly.
It is outrageous.