The only problem I have with that is that one end of the power socket doesn't expose any pins, or perhaps only one. How would you do a continuity test on this?
In the photo that accompanies your text, the cable harness has two wires, one of them red, the other not visible. The round connector with the two nuts threaded onto its body appears to be a "barrel" connector, commonly used to interface with "wall-wart" power supply cables. There should be a center pin visible inside the barrel connector. The outside of the mating barrel plug (connected to the wall-wart) mates with an inside conductive sleeve while a hole in center of the mating barrel plug mates with the lone center pin inside the barrel connector.
There are several approaches to measuring continuity of the complete cable harness, including the barrel connector with the two nuts attached. Firstly, identify which pin of the two-pin connector the red wire is connected to. From the photo, it appears that would be the left pin in the red connector housing. Using a small alligator clip or, better yet, a mating female socket pin, place one test lead of your multimeter on the pin identified as being connected to the red wire. Make sure you do not "short" the alligator clip or the female socket pin to the adjacent male pin!
Attach a small alligator clip to the other test lead of your multimeter and use it to hold a short length of conducting bare wire. A paper clip often works well for this. Now, with the ohmmeter set to a low ohms range, or to continuity mode, touch the short length of conducting wire to the center pin of the barrel connector. There should be continuity if the red wire is associated with the center pin of the barrel connector. If no continuity, move the short length of conducting wire to contact the inside of the barrel connector. If there is continuity there, you have identified one end of the red wire connection.
Now move the alligator clip attached to the male pin inside the red plastic connector housing to the other male pin. Repeat the continuity tests to the center pin and inside conductive shell of the barrel connector to verify that you have successfully identified the two connections.
Often it is not absolutely necessary to include the connector pin structure in continuity tests. Usually continuity problems are caused by broken wires or bad connections, especially poor solder connections which can be real bitch kitties to find. So eliminate the wires as potential problems sources by performing end-to-end continuity tests on just the wires. This is most easily and quickly done with needle probes that pierce the insulation to make contact with the wire underneath. Needle probes used to be "standard" equipment in the 20th Century, and steel phonograph needles were readily available sources of replacement needles. Today, you might consider using a sewing-machine needle instead. These are easily gripped with alligator clips. In performing the continuity checks, you might as well make sketches that identifies how all the wiring is connected. This will come in handy when the real troubleshooting begins.
"From the connector pins on the board". Are you referring to the gold ribbon cable?
"to the individual conductors on the other end of the ribbon cable". The ribbon cable's other end goes under the board. Just want to clarify you are suggesting here to remove the board and test where this ribbon cable leads to or somewhere else?
Yes. The gold ribbon cable does not appear to be connected in the photo. If it is in fact securely inserted in the mating socket, it wouldn't hurt to verify there is continuity from the circuit board connections, the pins on the connector, all the way to the other end of the ribbon cable. This type of flexible, "printed circuit" ribbon cable is not very prone to failure, but the connections to either end can be problematical.
Troubleshooting electronics is a skilled art that is helped immensely by knowledge of electrical circuit theory as well as electronic circuit theory. Back in the day, it was quite common for DIY TV "repairmen" to replace every tube, one by one, in a television set until the got the set working again. Often the "customer" paid for a whole bunch of new tubes that didn't need replacing, if only the DIY "repairmen" knew WTF they were doing. We called this "shotgun maintenance" back in the day, and it was (and still is) common in DIY automotive "troubleshooting" with lots of needless "repairs" being performed by ill-trained or non-trained shade-tree, weekend mechanics. Please don't place yourself in that category. Take it easy. Learn the basics and proceed with some knowledge of what you are trying to do. Best of luck to you!