That's exactly how we made my high-school newspaper. Then about 1987, when I was a junior, the first day of Journalism class we walked into the room and discovered our world had changed forever. There were now two Macintosh computers (one with a gigantic 20MB hard drive).

We'd stepped into the future.

The future back then was you printed the proofs on a LaserWriter and then took those to the dark room. And maybe had a halftone screen if your school was fancy.

What's not clear to me is what happens after. It looks to me like photographing the end result would result in a lot of visible artifacts from the glue and bits of paper? (I watched the video without audio, so maybe they explained this.)

Maybe this is just the process used for designing the layout, which is then used to re-flow and re-print all the "fragments" so they can be remounted and photographed cleanly?

The board shown was camera-ready and would be photographed like that, though maybe after cleaning off some glue with a rubber cement eraser. It's a monochrome camera that will only pick up blacks and reds, the lines are drawn with blue pencil so they won't be visible. It will photograph cleanly.

So the process would not work for full-colour magazines like MacUser?

Same process, you need to repeat it for each color that you would want printed.

This video shows a bit of the process such as the Rubylith sheets used to create masks for each color: https://www.youtube.com/watch?v=fxddg3dNKgw