2phar wrote:
I'm curious if the controller ICs in other LCD monitors typically run

this hot (pretty much too hot to touch) and without heatsinks.

I have seen ICs run hot as you describe and they seem designed this way, but you should check all its supply voltages. If you find its datasheet it will tell you the operating temperature ranges so you know if it is getting too hot.

The on-off thermal stress over time may have broken some really small solder or PCB trace and it is failing under heat. Fixing that may be difficult if not impossible and a practical solution to be tested is to install a small fan on the cage with a small heatsink on the IC, with luck it may allow it to run enough time to make it useable.