I measured the 12V supply line to the TCON and that read ok. Inspecting the board, I found an open SMD fuse. The supply line did not measure completely short, so I tried my luck and soldered a normal fuse on top of it for a quick check. (Don't mind the funky soldering of some parts, I've used this board for SMD soldering exercises...)
After switching on the TV, I heard a popping noise. On the bottom side of the board, I found a blown ceramic decoupling capacitor. That must have been the culprit all along. To find its original value, I looked up the data sheet of the DC-DC converter chip it was connected to and tried again with a replacement, but the board was still shorted and pulled the 12V supply line down. This cap may have killed some other voltage regulator upstream in the supply chain. On the internet I found one picture in a forum with the exact same problem. This might be a common failure.
I found a new TCON on ebay and that fixed the TV. As a precaution, I
backed up this apparently undersized capacitor with a bigger one, taken
from the old board. The "big" guy (still microscopic) has a lower ESR
and should be able to handle the current pulses better.
My theory is that the DC-DC converter is pulsing heavily and the little
ceramic (1µF) is the closest to the chip. So it is taking most of the
pulses and that is getting too much after a certain runtime. My set had a
whopping 12000 hours since 2009 (in Jan 2015).
The Sony is now in my living room and I love its image quality.
Generally, Sony TVs seem to be very solidly built and I found only
highest quality capacitors on the boards. No junk anywhere.
The TCON's number is 46NN_MB4C4LV0.7. The metal cover has the number LJ63-06901A
(FROM:alpengeist-tvrepair.blogspot.com)