Some TVs use a standby supply that runs all the time and simply feeds the 5V
and enough 12V to kick the relay in. Once turned on these supplies are
"overridden" by a run supply, sometimes from the horizontal.

If you've actually got the 12V, but no input, something else that is supposed
to run from the source that feeds that regulator is down, and probably causing
your symptom.

Usually they use blocking diodes though, and a shorted one of those was
probably the original cause of the problem. Best I can do with the info
provided.

Good Luck.

JURB