Those DMMs with audible continuity testers will sound when the resistance measured is below some threshhold, but not always that close to zero ohms. In same cases, you get the beep but the resistance is still too high for things to work.
"The one-wire continuity problem was one I had faced in the past, when I was helping a friend fixing up a very old two-story house."
Two story, one wire, but I ahve three stories. Does that mean I have two wires? Actually, I have 12 Ethernet lines with eight wires each. To locate and document the connections, I used a DMM after making a test cable that shorted the two middle wires together. Then I just had to run around with the sorting cable to each Ethernet jack and listen for hte beep in the wiring closer where all Ehternet lines come out of the wall.
It is not uncommon for someone to take an axe to the main cable. These are the same as a telephone switch trunk, about 2 inches or 5cm in diameter. Most modern installations use telephone 50 pair wire code. The old stuff is cotton covered.
With over 1000 wires in the bundle, it is usually easier to replace with the color coded stuff. Otherwise it is toneing the lines out against a common.
I keep thinking there should be some sort of device which puts a different frequency on each line, then tells what connects to what.
Well, you can test 8 cables at a time with a 'cable toner', such as ones made by Aska or Fluke. They are meant for coax, but I don't see why you couldn't use them on other cables, as long as you have a common ground.