Giving the OP every benefit of the doubt, here's my guess at what actually happened: They arrived at the gate 35 minutes before departure, the boarding passes were rejected when the agent scanned them, the OP was told to get in line to talk to another agent, it was a long line, boarding completed and the flight closed while the OP was still in line, and when they finally reached the head of the line were told nothing could be done and the agent had to go work another flight. If that's how it went down, then I assume someone or something had offloaded them from the flight prior to boarding, possibly because of a very tight international to domestic connection, as suggested above. I would not be surprised if the AAgent was snippy and gave them a "talk to the hand" response, but we've all had that.

And if that's what happened, most of the commentary above could have been avoided if the OP had just said that. Misconnects happen all the time. Sounds like they got to their destination the same day, albeit late, and got some miles in compensation. I'm sure it was very stressful with the baby, but I don't know what more AA could do.

I don't get the "forced to book new flights to Washington later that night" comment. Did they have to buy new tickets, or did AA just rebook them? I can't imagine they had to shell out cash to complete the journey. If so, then there is way more to the story than we've been told. But I doubt it.