It's plausible that our current 3-D printing design workflow has grown up in the computationally intensive way that it did
because all that computation was available; it's easier to just use a least-common-denominator but bulky file format like STL than to worry about the incompatibilities that would result from using more expressive files.
People have been doing CAD/CAM since the 01950s. Boeing started using CNC in 01958 on IBM 704s, and MIT's Servomechanisms Lab (working with the Aircraft Industries Association: https://web.archive.org/web/20090226211027/http://ied.unipr....) sent out CNC ashtrays to newspaper reporters in 01959: https://en.wikipedia.org/wiki/History_of_numerical_control#C.... Pierre Bézier started writing UNISURF in 01968 at Renault, who was using it to design car bodies by 01975. The Utah Teapot was created in 01975, and it consists of nine Bézier patches; you could print the whole dataset on a business card: https://web.archive.org/web/20141120132346/http://www.sjbake...
The IBM 704 was a vacuum-tube machine that could carry out 12000 floating-point additions per second and had a failure about once every 8 hours https://en.wikipedia.org/wiki/IBM_704. The Intel 8008 (not 8088, not 8080, 8008) that came out in 01972 could carry out over 79000 8-bit integer additions per second, which is about the same speed. But much faster computers were already available, such as the PDP-8, in wide use for real-time control, and they very rapidly became much cheaper. Any computation MIT's Servomechanisms Lab could do in the 50s was doable by hobbyists by the 80s.
The reason 3-D printers mostly use stepper motors is that they don't require closed-loop feedback control. 2-D printers from the 01970s used stepper motors for the same reason. They were accessible to hobbyists; in the 80s I had a Heathkit printer someone had built from a kit in the 70s.
If you wanted to print Frank Sinatra's face on a toilet, I think you'd probably want at least a 64×64 heightfield to get a recognizable Sinatra; 256×256 would be better than the line-printer pictures we were doing. 8 bits per heightfield point would be 65 kilobytes, which would fit on the floppy disks we were using at the time. This would have been totally feasible, though digitizing Frank Sinatra would have been a nontrivial project, quite aside from printing him.
So I don't think computation was the limiting factor.
Your "basically banging into the object with a very crude sensor, so there's no generation of geometry at all" is called a "pantograph" and it has been a common way to copy three-dimensional objects and engrave letters with a milling machine for 180 years: https://en.wikipedia.org/wiki/Pantograph#Sculpture_and_minti...