My point about APT is that a "beefy machine" in 01959 was the embedded controller in your keyboard by 01983; it wasn't a beefy machine any more. The PDP-8 was indeed pretty common for CNC control in the late 01960s, and it could run about 333000 12-bit instructions per second, which is about half as fast as the original IBM PC. So, yeah, a machine like that was exactly a home PC by the mid-80s. For real-time control of the 3-D printer, you can get by with less.
There were three big problems for manipulating 3-D models for home hobbyists in the 70s and 80s.
One was computational speed: for games, you need to render the 3-D graphics fast enough to provide an illusion of immersion, and with a few hundred thousand instructions per second (and no multiplier) you were limited to a handful of polygons. Like, typically about 20. See the Apple ][ FS1 flight simulator https://youtu.be/lC4YLMLar5I?&t=93, the 01983 Star wars arcade game (at 2:02), Battlezone (2:14), and Ian Bell and David Braban's Elite from 01984, which was successfully ported from the 2MHz 6502 in the BBC Micro (2:53, but also most of the rest of the hour of the video) to the Z80-based ZX Spectrum (3.5MHz but noticeably slower than the 6502; see https://www.youtube.com/watch?v=Ov4OAteeGWs).
For producing G-code, though, you don't need to be able to handle all the 3-D geometry in 50 milliseconds. You just need to be able to handle it overnight. That's a million times longer, so you can do a million times as much computation on the same hardware. You can't handle a million times as many polygons, because you don't have enough storage space, but you can represent geometry in more expressive ways, like Bézier patches, nine of which made up the Utah Teapot I mentioned in https://news.ycombinator.com/item?id=42080437, or parametric or implicit equations, solids of revolution, CSG, etc.
You do need some kind of user interface for seeing what you're designing that doesn't require waiting overnight to see the results, which I think is what you mean by "a meaningful way that would really be approachable to a general hobbyist". But it's possible I have a different concept of general hobbyists than you do; as I remember it, home computer hobbyists in the 70s were constantly writing things like
2120FORQ=1TO4:IFLEFT$(R$(Q),1)=O$THENRC=Q:ST=ST+2*Q:DX=DX-2*Q:NEXTQ
But the user-interface issue gets us to the second problem hobbyists had with 3D in the 70s and 80s: display hardware. The limited displays of the time were hopelessly inadequate for displaying realistic 3-D. The Star Wars arcade cabinet mentioned above used a vector CRT in order to be able to do high-resolution wireframe, because framebuffers were far too small. With color palettes of 2–16 colors, even Lambertian flat shading was nearly out of reach until the late 80s.
Again, though, this is much more of a problem for games than for 3-D printing.
My first experience with CAD was on an IBM PC-XT with a 4.7MHz 8088, roughly five times faster than the BBC Micro Elite is running on above (https://netlib.org/performance/html/dhrystone.data.col0.html). I was using AutoCAD, a version whose 3D functionality was inadequate for any real use, and the machines had two video cards—a text-only MDA (basically just a character generator and some RAM) and a 320×200 CGA, on which I could see the actual drawing. Redrawing the whole CGA was slow enough that you couldn't do it after every drawing or erasing operation, so erasing a line or arc was instant, but would leave holes through the other lines it had crossed. Until you issued the REDRAW command, which took typically 1–5 seconds for the fairly simple mechanical drawings I was doing. (Remember that 2-D side-scrolling games were impossible on things like the CGA because you just didn't have enough bandwidth to the video card to redraw the whole screen in a single frame.) Zooming in and out also suffered from a similar delay, but was very necessary because of the low-resolution screen.
Once I finished the drawing, I would plot it out on a CalComp pen plotter, which was basically a 3-D printer without the Z-axis. This had enormously higher resolution than the CGA because it didn't have to contain RAM to hold all the lines it had drawn; the paper would remember them. Some hobbyists did have HP-GL pen plotters during this period of time, but it wasn't very common. I had one at home in 01990. My dad used it to make multicolored Christmas cards.
This sort of approach allows you to provide a real-time interactive user interface for modifying geometry even on a computer that is far too slow to rerender all the geometry every screen frame. It would have worked just as well for 3-D as it did for 2-D, which is to say, it would have been clumsy but adequate.
But the third problem was, as you said, the knowledge. Because the 3-D printers didn't exist, hobbyists didn't know the algorithms, they didn't know the vector algebra that underlay them, they didn't have the software, they didn't have BBSes full of 3-D models to download, etc. That would have developed, but not overnight.