I found that if I power cycled the phone, I had around 20 seconds of usage before the same behaviour started again and I could keep doing this process.
I tried factory reset, and found that the screen worked perfectly for bootup and would work continually until any button was pressed - then the same 'screen switching off issue'. I could get a flicker of what was on the screen, by pressing the power button - but then it quickly gracefully went off.
The screen itself appears to be functional, and no odd pixels. Booting (and using) the bootloader / recovery works exactly as expected, without the screen turning off - for an unlimited amount of time.
I then tried flashing Cyanogenmod, but gave the identical result. Now reverted back to stock.