Well, I had a major setback trying to diagnose the repair. Last night I set it on the dining room table, face down, and this morning I did the flashlight and audio test again before even taking the back cover off, just to confirm the video and audio were still working. But after about 30 minutes, the TV started working again. All the backlights came on, and the picture is perfect. This is the behavior the owner reported - it might get backlight at any time, or it might not.
I measured 267V between BD9101 and J858, so that's the benchmark in case it dies again.
So I don't know how you diagnose anything when it's working properly. We tried tapping on every part and connector on the power board, but not even a flicker in the lights.
Both sides of the power board look perfect - not a hint of a bad solder, no domed capacitors, no brown marks. The connector to the LEDs (CLN802) has 7 black wires and one blue wire, and a legend that doesn't seem to match the connector.
In any case, I will let it cool down, and see how it powers up. I guess I could remove and test all the electrolytics, but there are a lot of them. Same for the diodes. But it seems it could be anything. Major bummer that it's working. :-)