Chosen Solution
Removed power supply board (BN44-00423A) and re-soldered all heat-related connections. The voltages on the board seem fine as printed on the board (5v and 12.8v). Was able to do a factory reset before the last shut down, no change in behavior. Could this be the mainboard shutting down the unit? Is there a good way to tell what is telling the board to shut down after 30 seconds?
@dorkygrin pretty much any ad board could be the cause of that. Unplug the main board from the power board and see if your backlight come on. That way you know that at least they get power fro the board and if the TV stays on then the likelihood of the issue originating on the main board is great. Measure all the voltages on the connector from power board to main board and let us know what you find. It will also help if you can post some good pictures of your boards with your question. That way we can see what you see. Adding images to an existing question