If the logic board is working fine, check resistor R20 on the Analog board. It should be 100K. If this one is open, you get an all white screen and the brightness cut-off doesn't work. I've had this issue with one of my SE's a few weeks ago.

If you haven't recapped the PSU and Analog board, it's definately recommanded as most capacitors are degraded due to age as they are over 22 years old now.