Cooling a component causes it to shrink and possibly deform enough to make/break a cracked solder connection. Cooling a semiconductor junction can help spot leakage currents, noise, voltage offsets, etc. that vary with temperature.
In the case of BGAs, it can help find a cracked solder ball or poor joint under the package.
My next step on troubleshooting this beast would be to scope all the power supplies for noise (bad caps) and then try a reflow of the main PCB. I really don't think this is a panel issue.
Finding a used board on eBay is always an option. Have to decide if the monitor is worth the $80-$100 gamble, or find a seller who accepts returns. If you were able to troubleshoot the boards to component level, it would likely be one of the big custom ASICs anyway, which would be nearly impossible to get replacements for.