Can a faulty board, let's say burned in a "defrost" mode, cause the entire refrigerator to stop working? It seems to me that the compressor won't run while the unit is in defrost mode and if mine is stuck in permanent defrost, it may explain why the compressor won't start. I have already replaced the compressor start relay with no change.

That board controls the entire operation of the refrigerator. If anything is wrong you need to replace it.