Just look at the little one! :-)
Standby voltage was ok. So I focused on the mainboard's DC-DC converters. They all measured ok. Was this probably a CPU or EEPROM problem? Hard to tell and not fixable for me.
I then remembered a YouTube video from the infamous EEVBlog, where Dave fiddled with a processor stuck in reset mode. So, just for fun, I located the reset pin at the CPU and put the multi-meter probe against it. The capacity of the meter was enough to pull the pin low for long enough to make the TV reboot. And lo and behold, something happened and the backlight turned on. Sometimes. Most of the time nothing worked. Once the set even booted fully, but didn't stay up long.
Suddenly, I heard a buzzing noise. The 3.3V provided by the main DC-DC converter chip U702 was wildly oscillating. It stopped freaking around when I pressed my finger on the chip, and the TV booted once I reset the CPU! So, this regulator did not provide a stable enough voltage during the boot process. Afterwards it looked as if everything was ok. Obscure failure that one.
Ok, I ordered 5 pieces of the TPS54319 chip from a seller on Aliexpress.com. Those QFN16 package chips are tiny and very difficult to solder. I fried the first one and it took three attempts until the thing sat aligned with the solder pads. Would you believe that this breadcrumb is capable of 3A output current? I find that quite astonishing.
The TV worked! It has a surprisingly good image, once the PC mode is selected and all the silly image enhancers are bypassed. Frau Johansson radiated into the room as good as it gets.