Unless I'm missing something it doesn't really matter when they got to the gate. If the kid decided to soil itself a few minutes before take off, then it doesn't matter how early they got there.
20 minutes before the plane is supposed to take off is way to close though. Kinda hard to take sides without knowing more. Maybe they had good reason for being late, such as a massive check-in line, late cab, etc. I've experienced d-bag people working at the airport so I have sympathy if that was the case.
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