My first thought was the PSU, but it's an external one and replacing it with a bench PSU doesn't help.
I opened it up and found debug headers for the serial flash and EEPROM next to the Weltrend controller. A logic analyzer shows activity on these lines every time it resets. After looking around on the Wiki and forums I tried the unbrick method by holding SDA high, but that just caused it not to do anything except flash the standby LED (looks like it might be an error pattern: 3 blinks, pause, 2 blinks).
Does anyone know how to fix this? Or at least where the problem might be?
But in general, here are a lot of other reasons to behave TV like yours, e.g. firmware upgrade failure, which is un-repairable ATM.