The number 1 (or number 3) line pilot on the seniority list at AA, who has 39 years at AA, has no more non-rev priority than the most junior AA employee with travel passes, even if that person was a 20 year recent hire as an assistant to the toilet cleaner some outstation. That toilet cleaner lists himself for a flight with one standby seat available 1 minute before the number 1 B777 CA does, tidy bowl guy gets the seat.
The few management pilots with upper level status may have F-Class rights, but as I mentioned, they are very few and it has little to do with actual seniority.
If the #3 guy isn't one of the top management dogs, the wifey is pulling your chain.