When did they finally get rid of the green screens?

They haven't. That's why the system feels like it's stuck in the 80s.

About 1 in 8 of my AA flights comes with, “When you reach the gate please say your last name and seat number, our terminal isn’t scanning right now”

Websites are very modern interfaces to the Sabre system that was originally developed in the 1950s. The way it was described to me at the time I was working in the travel industry (20 years ago, yikes!): Sabre is essentially a command-line system, which means the API-like interfaces that have been grafted onto it over the decades are not nearly as stable and robust as we'd like them to be. I'm sure they've improved tremendously since I first used them, but the entire system is still based on very old technology that just wasn't meant to do what we're doing with it. Connect to it with an airline agent's terminal, though, and it's quick and reliable, just as COBOL-based banking systems are.