Thanks for the suggestion projectgus. At this point anything is possible. ;-)
I did some further testing on the board yesterday to check the chips connections to other points on the board.
Pin 1 and pin 4 of the J516T chip are connected together, and together connect to a zero ohm resistor (R394) (I think effectively a fuse), but it doesn't seem to connect to anything else on the board, which is really weird. The only thing they might connect to is a connection on the thin ribbon cables that run to the LCD panel, but I'm unable to probe those.
Pin 3 goes to one pin on the input connector CN2 (pin 3) (which I think is a power supply pin, but I'm yet to confirm) on the other end of the board and there is one bypass capacitor (C305) on the line near the input connector.
Pin 5 goes to three sequential pins on the input connector CN2 (pins 68,69,70) on the other end of the board and it also connected to 8 bypass capacitors (C304,C316,C323,C326,C333,C336,C343,C346) going to ground along the way.
Pin 2 is ground.
I've captured a high-resolution image of the board, which helps understand the configuration. The J516T chip is removed in this image.

Unfortunately, it seems the board is multi-layer, so this top view doesn't tell the full story.
But... given it seems there are only bypass capacitors connected to the J516T chip, to me it seems likely to me that this is a Transient Voltage Suppression (TVS) chip.
Given it's only one of the diodes (pin 1) that is malfunctioning on the bad J516T chip (which should have a forward voltage of 0.7V) I might just solder the chip back with pin 1 disconnected, and see if this allows the LCD panel to behave correctly. I could solder a small external signal diode between pins 1 and 2 to replace the missing diode in J516T, but I wonder if that is absolutely necessary. I was thinking of drawing up a simple schematic to illustrate what the circuit looks like.
But as you can see from the circuit board image, the J516T chip is the only chip on the board. Everything else is just capacitors and resistors.
All comments, ideas and theories very welcome! Thanks for reading this far!