Of course AA has access to TIMATIC. It's an IATA database and it exists for IATA members. But, AA is free to impose stricter requirements than those imposed by the countries you are visiting. Same thing as many of the European LCC's which require passports for travel where the law does not require a passport.

Showing up at JFK with the printouts from TIMATIC won't necessarily help you.

You are risking your ticket and whatever it is that you want to get done abroad by not renewing your .passport. As you are going to have to get it done anyway, might as well get it done now. As a NYC resident with a Passport Office in the City, you could obtain a passport for travel on Monday night by showing up on Monday AM. A hassle for sure, but doable.