IMHO - the Z80 is probably not the optimum starting point. Its close cousin, the 8080 started out as a TTL cpu built into early Datapoint terminals. Intel took the Datapoint logic design and ISA and integrated it into LSI silicon. Anything a Z80 can do, an 8080 can do - albeit less efficiently.
With an 8080 equivalent running a serial character display terminal based on an oscilloscope CRT (1940s RADAR tech) you have an input/output device.
This leaves the main job of processing to another cpu, which could be 16-bit for arithmetic speed and efficiency. The late 70s, early 80s 8-bit machines were only underpowered because they were doing all of the video output using the same cpu. Separate computation from video generation and you get a much faster system.
8-bit cpus rarely needed an OS. They were really only capable of running single applications at a time. All an operating system does is separate hostile C code applications from each other. C is probably not the best starting point to reboot society using 8-bit systems.
Forth, or some derivative might be better. Charles Moore's original 1968 listings for Forth on an IBM 1130 are available from here: https://github.com/ForthHub/discussion/issues/63
Remember also that every mid-1970s microprocessor generally relied on a minicomputer (built from TTL) for its software and logic design. If you go back 10 years (1965) to the PDP-8 minicomputer, these were built from diode-transistor logic or DTL - made from discrete diodes, transistors, resistors and capacitors. This sort of technology could possibly be re-booted more easily for post-apocalypse society.
The original 12 bit PDP-8 contained 10,148 diodes, 1409 transistors, 5615 resistors, and 1674 capacitors. See- https://www.pdp8.net/straight8/functional_restore.shtml
Scale these figures by 1.33 and you have the approximate requirements for a 16-bit architecture.
Whilst over 50 years old, the PDP-8 could run BASIC at speeds not too dissimilar to the early 8-bit micros that appeared in 1976 - about 10 years later.
It used a modular construction - and if you did find yourself with an excess of diodes and transistors, the best approach might be to build a series of logic modules - loosely based on the 7400 series, but using DTL for simplicity. If you were to standardise on a footprint similar to a 40 pin DIP, you could probably recreate about 8 NAND gates in such a device.
Some years ago I looked at the NAND to Tetris cpu, and worked out a bitslice design based entirely on 2-input NANDs. Each bitslice needed 80 NANDs, so a 16-bit machine would need 1280 gates. Memory would be difficult, but something could be implemented using shift registers. You could of course revert back to storing charge on a CRT screen - which formed the basis of the 1K words of memory on the Manchester Baby machine of 1949 (Williams Tube).
Finally - never underestimate audio frequency generation, and storing signals as audio tones - something that cpus are good at. Possibly use a rotating magnetic drum for storage.
In the summer of 1984 - a friend and I, who both owned Sinclair ZX81s set up a 1-way data link between one machine and the other across our college dorms - using a FM transmitter bug and an FM radio receiver - over a distance of 300 feet.