What time did the app show your inbound flight arriving?

It seems to me that everyone you're interacting with thinks your inbound flight blocked in on time, which is why they're treating the outbound as a noshow. If your flight did block in on time and then your ability to disembark was encumbered by jetbridge issues or similar, that's something you need to highlight to AA, because their systems don't do a good job capturing those types of issues.