Hello, I’m Nathan, and I’ll be assisting you today. My livelihood depends on your positive feedback, so your satisfaction is my highest priority.
Sorry to hear you’re having trouble, Monty.
If it powers up and shuts back down, that isn't a power board issue. The caps likely were starting to fail, and contributing originally, just due to the age (2008 model).
Bad caps won't cause startup and then shutdown though, it just wouldn't start up in the first place.
The on/off cycling will be attributed to a bad main board, which comes down to failed integrated circuitry (bad eeprom). There is no quick/easy fix for that, it isn't a simple solder job, you'd actually replace the board.
That will be about a $70 part. It certainly can be swapped out, but honestly for a tv this age and size, you're probably better off replacing it entirely. You can get a brand new LED tv with a fresh warranty and better picture for $150-200. Repairing also won't extend the life of the tv significantly, you still have an aging power board / inverter that is likely to fail at any time now.
Depending how you'd like to proceed here though, I can help with part sources if you'd like to replace that, or recommend replacements if you prefer.