Finally figured out a brake cable solution. 1981 Special with drum brakes front and rear.
I ordered a Venhill Universal Kit. It didn't work. This was kit U01-4-101-BK. I sent it back. The inner cable was small; didn't make me confident about it's suitability as a brake cable. Parts weren't the right bits either.
Ordered a second Venhill Universal Kit.
U01-1-1-100-BK. It came with all of the correct bits, an adjuster that was the proper spec, and the inner cable was nice and robust.
I cut the new housing a bit shorter than the OEM as the bars we are using require less cable length. I re-used the rubber boot but otherwise I used all new hardware from the Venhill kit. Incidentally, the adjuster is longer by about 8mm and the inside opening is smaller, giving the impression that the walls of the adjuster are a bit thicker. The old adjuster, thought it may be 40 years old, is most likely made of better metal.
Adjusted the brake actuators a bit so both shoes were working together as they should.
Soldered the new cable end; it's basically identical to the OEM cable end.
Cable works great; nice and smooth even before lubricating, even better after.
Here's a pic of the new cable. I'd highly recommend the Venhill kit after using this one.