A 35mm film projector was a machine with a handful of parts. If one broke, theaters could be back up within a matter of hours.
Are you kidding me? A 35mm projector has hundreds of precision engineered moving parts. If one of them breaks, you're not swapping it out with a replacement from Radio Shack. You might not even trust the pimply teenager running the projector (me) to touch any of the interior parts or gears.
You're going to the pros: Authorized service techs, or a specialized AV supplier who may have it in stock, but is definitely not open on the weekend when these things tend to break. And, the supplier may have have to order the replacement part from Japan or Europe.
If you're part of a chain that uses standard equipment (not a given, considering consolidation) you might be able to borrow parts or spares from your pal running the show at the multiplex across town.
Worst case: Because most theaters operate two sister machines for each film (a reel can only hold ~75 minutes of film max, so you have to switch over to the second machine halfway) you can technically swap reels on a single machine with a 5 minute intermission.
The 35mm projectors were professional-grade machines designed to last for years. Our theater used 25-year-old Italian projectors made by Cinemacannica which is still around today (https://www.cinemeccanica.eu/cinema-products/), and looks like it has moved into digital exhibition.
Some of the old 35mm machines are still around, usually at indie theaters. I spotted one a few years ago in Somerville Mass as described here: https://leanmedia.org/memories-movie-theater-film-projection...