For each number, I mark all cells where I know I can’t put it.
If one block, column or row has only one possible cell where a number can be, I put it there and then update all cells where I now know I can’t put it anymore.
This algorithm probably isn’t the fastest, but it’s easy to do manually and it has the advantage that you don’t need to backtrack.
If the controller port is connected to the same +5V rail as the CPU, wouldn’t the NES crash if it only got 4.6V or less?
If only the BlueRetro is affected, maybe something behaves like a resistor in series with it, for example a broken solder joint in the adapter or at the connector on the NES