[Fraud]

Ok so I'm a damn fool. Grounded it properly from both ends of the tan wire and the gauge does indeed go to empty, so the gauge works as expected and I'm even more confident I have continuity all the way to the sender.

So got the multimeter out again, and measuring resistance from the connector on the sending unit to a good earth. I was really just testing for continuity because my working theory was that a connection came loose in the sender and it wasn't grounding, but I got a reading...

Unfortunately I'm seeing a little over 1k ohms. Zeroed out the meter again to be sure...

So... Dead sending unit?