I'm going to have a go at this, but it is honestly hard to make sense of.
Historically¹, films were screened on shortplay² projectors, which could load only a single small reel at once (~15 minutes of playtime). To play a longer film, the cinema would set up two projectors, loaded with the first and second reel; when the first reel was exhausted they'd switch over to the other projector. While that was playing, the first reel had to be unthreaded from its projector and rewound, and the third reel prepared and threaded into the projector in its place -- a fiddly and labour-intensive process. On top of this, someone needed to be constantly tuning the projector lamp, and if you didn't have one of those newfangled electric-motor projector you also needed someone cranking the projector. A feature film presentation could involve 4-6 people working in the projection booth for the entire duration of the film.
In the 1950s, longplay systems started to arrive, capable of displaying an entire feature film in one go; this required someone to splice together the various reels before the film started (not just the feature itself, but also advertisements and other front- and back-matter added by the cinema), but once assembled the projector would run more or less without attendance. Self-threading projectors caught on in the 60s and meant the process of loading the film into the projector itself was now much faster and simpler, too.
This, and other changes (like xenon lamps replacing carbon ones), meant that the role of a projectionist became more about setting up the projector and reel before each film and doing routine maintenance on the equipment during downtime, and often a single projectionist could manage multiple theatres at once.
What makes the quoted passage confusing is that the job of splicing together shortplay reels into longplay reels or platters is a job that was introduced in this period, and stuck around more or less until digital film replaced optical.
Given that, my best guess is that it wasn't about the specific job he was doing, but more about the near-total collapse in demand for projectionists between the 60s and 80s. He likely was working at a smaller chain with older equipment, but as they swapped in more modern gear, the number of projectionists needed would drop by ~90%, and the work they would be doing would skew more towards technical maintenance work. With that happening, the most junior projectionists would probably get the axe first, including Tyler.
¹ Early to mid 1900s
² I'm not a film buff and may have some terms of art wrong here