I work in the travel industry as a programmer god only knows for how much longer - I can tell you that Sabre and other GDS's are only used if you go through a travel agent or use some online reservation systems. If you book through the airline's systems or on online reservation systems they likely use the airline's systems to track travel instead of GDS since the GDS wants to take a big cut of every ticket sale.
And obviously only legacy travel companies like Hertz and Mariott integrate with GDS's, new travel companies Uber and Airbnb likely don't have any relationship with Sabre.
I used to work for Sabre, they do a lot more than just bookings, they provide all the software to run an airline basically. I worked in the area for scheduling air crew.
But also crossed over into all the flight tracking. Their systems hold a lot of data, though the airlines own and host that data in secure facilities. They claimed to employ the most PhDs at one time lots of operations research.
Also claimed to invent database transactions for the problem of people trying to book the same seat on airplanes at the same time early 60s I believe. As an airline you'll be able to buy an inventory management system from Amadeus, a pricing system from Sabre, a support system from TravelSky, and a website from Travelport. The best example I can think of is American. They have Amadeus running their international website. Their ticketing system is internal.
