I was on this flight and witnessed the entire incident, both on the plane and at the gate (where the video was filmed, after everyone was removed from the plane.)
I felt that this was a no-win situation, caused by bad policy. While I agree that (in theory) 2 year olds should wear a mask, in practice it is not always possible. The mother was not argumentative, she was just asking for a solution.
The AA (PSA) employees acted calmly and professionally, especially the Captain and the (ground) managers.
Could the FA have just "let it slide"? Absolutely. But she chose to strictly enforce a company policy, and that is her prerogative.
Ultimately they got the family home later that day, but a little more flexibility in policy or enforcement could have prevented this entire unfortunate situation.
On a side note, after the passengers (minus the family) boarded the 2nd time, there was another issue:
Prior to door closure, the same FA was walking down the aisle checking seat belts etc. She asked (politely) a man to put his bag completely under the seat in front.
A few passengers started giving her a hard time and made inappropriate comments. The FA was visibly upset about this and went to the forward galley, and the ground manager was called to the aircraft a 2nd time. The manager spoke to the passengers in the back, and then to the Captain.
Then the Captain got on the PA and made an announcement, stating that all passengers must comply with crew instructions, and if there was anyone with an issue, they needed to get off the jet now. He said that if there were any problems in flight he would turn the plane around, and we would return to Charlotte. He said he understood the concerns about the mask issue, and would be available to talk to any passengers that wanted to, once we arrived in Manchester.
There were no further issues and we arrived about 45 minutes late.