I worked as a production artist (someone who takes a designer's idea and turns it into somethng that can actually be produced, hence the name) off and on for 15 years, starting in the early 1970s and ending about 1991. I worked for an ad agency, typesetter, publisher, and finally in the art department of a flexo plate maker. When I first started, we used drafting boards, T-squares, X-acto knives, and rubber cement. Type was of the hot variety - photo type was only for display type, not body copy. In the late 80s I was trained on two SOTA production machines: a DuPont VASTER system (used both vector and raster graphics), and later a Context system that ran on Sun workstations.
The typesetter your parents bought was likely either a Compugraphic or A/M (Adressograph Multigraph) - in the pre-DTP days, they seemed to have most of the market for phototypsetting systems (at least for smaller businesses). I worked with both, and your description would fit either of them.
I guess I would somewhat dispute your assertion that layouts (or paste-ups/mechanicals as they're more properly called) took "forever" or that it was a "very long" process. Using the "Thin and Pretty" layout in the article as an example, the last place I worked we probably would have budgeted an hour for the job - typesetting, mechanical, camera work, and (maybe) printing plates (all done in-house). I don't think that's terribly out of line with the time it would take using DTP. (That doesn't include design time!)
When we were first making the switch from the old manual operation to digital, we often found it took longer to do the work using DTP; it wasn't unusual to hear an artist say "Screw it" and go back to his light table to finish a job. But that was to be expected - unfortunately, a number of my co-workers had little or no computer experience. Over time that ended, although the company did keep two full-time artists on staff who did nearly everything the "old" way until about 2000. The huge advantage of DTP, especially early on, was that changes/additions/corrections were so much easier. And in our case, it was considerably easier to actually find artwork on the computer. All our paste-ups were done on #20 illustration board (for dimensional stability) and they were filed in row after row after row of architectural file cabinets. Unfortunately, no one had ever devised a good filing system, so it would often take 2 or 3 or 4 times as long to find the freakin' art you needed to make a change on as it took to actually make the change...