When I worked at GCA on their wafer steppers and Wafertrac systems, many of the control boards were wire wrapped, even as production boards. They had all kinds of production errors and reliability problems, some from tin whiskers that would grow and cause shorts. The field service people know whick boards were the most likely to fail so they kept a supply with them. Management didn;t want to spend the money to design PCBs, GCA went out of business.
In the late '60s and early '70s I served aboard US Navy vessels that had commercial mainframe computers with wire-wrapped backplanes and individual transistor-based pcbs for each bit in the registers. We went from a Bunker-Remo computer with a 333 Kc clock and 2K of core memory, doing integral calculus, for a satellite navigation system in the '60s; to a Univac machine with a 50 NSec read/write cycle replacing the navigation computer and several others, in the '70s. The Bunker-Remo utilized 26 AWG wirewrap pins and wires, the Univac had 30 AWG pins and wires. The Bunker-Remo had in excess of 6K pins and over 30K wires, the Univac had more like 10K pins and 50K wires. To make things more interesting, the original backplane wiring was all blue wires and field changes were all done in red wires. Wire tracing was done with crochet hooks and all repairs/field changes were done by hand with wrap/unwrap tools.
I must admit that my experience of wirewrap was one of frustration and despair. I used it on my final year project at university and had a terrible experience. Only one joint in six was any good and invariably when I got one good joint its counterpart at the other end just would not wrap at all and each time I tried to wrap the wire again it would get shorter and shorter until I had to take the entire wire off - good joint included. Solder is by far a more reliable joint. Heck, even a solderless breadboard (they didn't exist at the time or were horrendously expensive, I don't remember which) is a more reliable method of connection.
I was interested in the fact it was used on the Apollo guidance computer. I can only presume that solder would get brittle in the cold or the joint be shaken apart by the vibration on take-off, in which case, a wire-wrap joint would probably make more sense. So it's perhaps a case of 'horses for courses'. I certainly wish I hadn't used it in my final year project. I'm not saying it's responsible for my poor degree but it was certainly a factor!
I worked at AT&T/Bell Labs starting in the late 70s on switching systems, and we used wire wrap for production and prototype backplanes, as well as prototype circuit boards (circuit packs as the Bell System called them). Since hardly anything ever gets retired out of the legacy phone system, I'm pretty sure the stuff we shipped 40 years ago is still in service. We had fully automated wire wrapping machines; we only did manual wire wrapping to make changes. It was always fun when the wire you had to remove was on the bottom of the pin, and so you had to remove the two or three wires above it, and then those wires had to be replaced, and they were at the bottom at the other end, etc.
In late 1970 I buult a system of about 100 Transistor Transistor Logic (TTL) using hand wire wrap like you describe. In early 1971 I was working in Ann Arbor and using software developed at the University of Michigan and full automatic wirewrap machines at Northern Telecom. I learned then your story about the reliability of wirewrap. We the used it in early protoypes of the computer that controls the engine in the car. The wiring survived driving on normal roads. A lot has changed in 46 years!