There are a good handful of stories here about folks getting screwed (after boarding the UA flight IIRC, when the GAs were running a check on the tickets or something) because AA hadn’t processed something properly. They had ticket numbers and everything I think but still got shafted.

When my my parents got rebooked by UA onto an AA flight due to a mechanical, everything went fine, I even got them each LFBUs at OLCI But in my paranoid state, I probably called twice and made sure to get the ticket numbers. If they missed *that* flight, they would have missed the family cruise.