>my "repair" skills would be limited to swapping out the whole thing or using some kind of robotic apparatus to gear down my movements.
Mine too! I never talked to anyone who did repair as a job, but my impression is that most repair is swapping out broken parts for working ones from other cannibalized movements. In theory you could fab new parts, of course, but that would be beyond my skills and experience. Think of it like how people build computers: they replace components and make sure it all works together.
I think for larger clocks (like the ones repaired by John in S-Town[1]), hand-created components were far more common. They were developed as objects that would be maintained and watched by a large community and less precision was manageable.