Wire wrapping was used in telephone central offices long before it was used for the general purpose electronics.
I used wire wrapping in the early 1980s. By then it was mature, with a body of knowledge from decades of experience. There was a large, detailed set of rules to make reliable connections. The pin and wire plating, wire gauge, insulation type and thickness, etc were precisely specified and standardized. If you followed the specified techniques, the result would be very reliable. Not quick, not easy, not easy to debug, not inexpensive, but very reliable.
For my hobby projects I had one of the inexpensive double-ended hand wrap tools. One end was for unwrapping, the other for wrapping, with a wire stripping groove in the middle. I think that most hobbyists of the era had that tool and a spool of the blue insulated silver-plated 30 AWG wire. A hobby alternative was a soft insulation wire intended to be used without stripping, with the square post cutting through the insulation, but that wasn't considered as reliable.
The setup I used for large boards was typical for commercial prototyping. There was wire storage, and the X-Y assembly machine. Wire storage was a panel with storage tubes for pre-cut, pre-stripped wires. Each tube held a different length, and the pre-cut wires had insulation color coded for their length. (Hmmm, DigiKey still sells the wire packs. $9.50 for 50 pieces of the shortest length, which is about what it cost "back in the day". For a typical 2000 wire board that's $400 for just the wire!) The assembly frame had two stepper motors that moved an L-shaped bracket. A tap on a foot pedal moved the bracket to the origin location and flashed the wire length on a LED display. A line-powered wrap tool ($$$) was used to spin the wire down. A second tap moved the bracket to the destination, which was always lower and/or left. That allowed for "Manhattan routing" (e.g. always horizontal then vertical) of the wire while the guide was slewing to the new location, and reduced the chance of damage from tapping the pedal a little early.