If the backlight is on a different board, you may find something
there.
It's possible that the new board had the same failure mode.

Normally, you'd expect the image to stay up even if the backlight
turns off, but who knows what the designer decided on that model.

Had one monitor that had a bad cap in the 5V. Supply.
The spikes on the 5V caused the controller chip to malfunction
and lock up.

Probe around with a digital storage scope to see what happens
on power up.

I neither own nor have access to any kind of scope.


that makes it difficult

I have seen descriptions of replacing the CFL backlights with white LED strips. That would make a nice project.

Mark