It is VERY possible for the xfmr to appear bad if there is capacitor acid on the board, at or near the terminals. The acid is conductive, the backlight power is high voltage and low current; any type of capacitor gunk in the circuit will simply short out the high voltage. I'd make sure the entire area around the transformer and related circuitry is absolutely clean before I'd assume the xfmr is bad.
As a test, you can temporarily connect just the primary (input) side of the xfmr to the circuit, leave the secondary (output) disconnected, and then check the voltage. That will tell you if the xfmr is good or not.
Charlie