Thanks for all the great answers you guys. I was leaning towards the
power supply issue. Another thing I notice is that when I initially
turn the TV on, a relay clicks on (most likely on the PSU), then as
the status LED goes to red, it clicks off. If I turn it on again soon
enough, I can see the backlight flash for a split second.

I've had good luck repairing PSUs with noticeable parts that have gone
bad (ie, electrolytic capacitors that are clearly not feeling well),
so I'll check this one out, too.

I'll check the outputs of the PSU to see if they are within ranges
that are standard (3.3, 5, 12, etc). Hopefully I can find my long T5
torx screwdriver since driver bits are too thick...