That was pretty damned close and you can visualize it on the ATC video you linked -
https://www.youtube.com/watch?v=9N1gDSZJ5s0
I did not fully understand it until I read the comments on that youtube video. The AA pilots appear to have missed their turn and they turned into the path of another aircraft taking off.
Thank God the ATC was on top of it!
They missed their hold. There are two parallel runways. They were supposed to hold prior to the Left runway but instead crossed over it to hold prior to the Right runway. It's a rookie mistake. They even passed signs that identified it as a runway.
Although the term “$h!t” might be improper protocol, it readily and clearly conveys a serious problem.