The defect showed as follows:
When plugged in, the blue power LED came on immediately and none of the other buttons reacted to touching. The device was totally unresponsive.
In this thread on Badcaps I learned that reflowing the main board revived some of the sets. Before that I did my voltage check routine and found nothing suspicious. The 23.5V were there and all the secondary voltages looked good as well.
The right chip under the heat sink was warm, the left was stone cold and its clock wasn't oscillating. It seemed as if the right main BGA chip would activate the left after booting.
So, my brand new reflow controller got something to prove its value. It is a clever device, which controls a normal el cheapo pizza oven precisely via a feedback sensor, which I taped directly on the BGA chip.
Before I ran the standard profile, I poured some liquid flux under the chip. I don't know if this had a part in the success, but I thought it wouldn't hurt.
And voilĂ ! The monitor ran flawlessly again. Hallo Frau Johansson!
During my research I came across an interesting software, which is able to reset the factory defaults in a monitor. It's called softMCCS and is available here: http://www.entechtaiwan.com/lib/softmccs.shtm
It identified my two monitors. First, the defect 2709 would be found, but DDC/CI was marked as "not supported". After the repair, it read "supported". So this is a nice tool to check whether the processor is working ok or something else is broken.
Rest in Peace
14 months later the error came back. A second reflow did not fix it. I also learned that the larger electrolytic capacitors did not like the oven heat. They were bloated afterwards. I'll remember that for my next reflow task.