An Observation
You and your niece are obviously upset.
This is a critical thing because it may cost her a job.
Yet your niece was denied checki-n and knows it was a male - yet did not get his name. I would have been pi$$ed and gotten his name - even if it were my fault for running late.
Also - you called the WeCare line - do not know who you talked to (they always give there name at the beginning of the call) - and them call them the We don't care line. I have found them to be very helpful.
You have been given good advice above - and if truth be known - I would not be surprised if you had not taken the previous poster's advice and printed off the 6:02 departure time.
Not trying to be pi$$y here - just think that you and the niece need to take on more responsibility.
And oh yes - I agree with the poster above. Let's hope youe niece does not tell the company she was late because she got there 3 minutes late for checkin. I would assume if she were late to the airport, she would be late to work.
Whew....now I feel like slapping my wife and beating the dog...