So, macho captain should throw off the customer who steps up to counter what he believes to be a wrong? Seems to me that's the employee attitude which creates these types of situations to begin with.
Airline employees need to understand that the burden of professionalism falls entirely on them in these situations. The paying passengers are customers. Customers aren't always right, but they should generally be treated as if they are. That's part of the job as an airline employee. Unfortunately, many of the regulations which are designed to keep air travel safe are instead abused by crew and employees on a power trip.
The photos/video of this incident seem to indicate that perhaps the captain was involved in helping to de-escalate the situation, but it's extremely difficult to tell what happened within the limited context provided by the photos/video. One thing is for sure, gate agents or crew members goading paying passengers are way out of line.
If there was a language barrier and inconsistent information provided to the passenger by airline employees, as mentioned by another poster, that's all the more reason why the airline should have been understanding.
With all of that being said, everyone has a bad day at work now and again. I don't think it's fair to airline employees or passengers that all of these altercations get recorded and blown up on social media. Yes, the United thing was inexcusable, but my opinion on United is that if you choose to fly them, you deserve whatever you get. This incident on AA (though still not my favorite airline) seems like something that perhaps shouldn't take on a life of its own.
JKG