We learned how to do it in 8th grade in shop class. The end result was a T-shirt or other item that we printed from the screens we made. We cut our screens manually with an Xacto knife, but also learned about photo emulsion screens.
Some years ago, I had a boxed software product made. I went to the box factory with a disk with the imagery, and sat down with the woman who made up the printing plates. She aligned the imagery to a box template, and I signed off approval. Connected to her Mac was a lithography plate maker the size of a car. That made the masters by photolithography.
I was invited to see the press. After putting on earplugs, we went into the press room. The press was a sheet-fed press about a hundred feet long, turning out about one large sheet per second. It was capable of something like fourteen layers, so it could do glossy and flat areas, intense saturated colors, shiny seals, and other special effects. Most of the time it ran with four to seven colors. That day it was turning out art prints, of the sort that appear in hotel rooms. When the box business was slow, they printed artworks.
Press setup and alignment was time-consuming. Cost about $4000 for the first box, $0.25 for the second and later boxes. The big advancement in presses in this century is that the alignment and color ink distribution settings are now automated, so there's a lot less trial and error at the beginning of each run. Older presses have a huge number of knobs and cranks.
That's the next step up in volume from silk-screening.
Electronic distribution is awesome, but there was something magical about seeing customers grabbing my boxed software off an end cap at Best Buy. Always fun if a customer was reading the box to walk up and ask if they had any questions. They'd ask "Oh, do you work here?" I'd just smile and say, "No, but I'm pretty familiar with that software."
I thought it worth mentioning that stickers were in the mix too. In the town I grew up in, there was a printing company that made them for Atari. I recall being brought to a "local industry expo" event as a kid, and I have a vivid memory of seeing heavy-duty vinyl (?) stickers for "Pole Position" at this company's stand. They would have been designed to be applied to the marquees and control panels of the upright cabinet.
Volume was roughly similar (thousands to low tens of thousands of cabinets).
Things differed from game to game over time. My Missile Command had a silkscreened control panel, but I think they went to overlays which were more durable and could be replaced.
I would imagine that on the big industrial printers that are using a metal mesh screen and thicker emulsion, it's maybe closer to 50? Usually in a print shop doing a big run, there'll be four or five screens made for each print layer so when one gets worn, it's replaced in the printing rack. It really is an artform.
This, it seems like a confidently incorrect answer, but I don't really understand why they'd act informed when they aren't.
Just like the screenprinted Atari cabinets, it's worth seeing the Spooky butter cabs in-person. While decals have come a long way in fidelity and application, there's a feeling of satisfaction when the image is part of the object, rather than on the object.
This is all familiar to people above a certain age. Younger folks who grew up without Sunday color comics might still be able to find a slightly blurry plus sign on the bottom of a paper milk carton.