I'd be on the look out for a weak solder joint, it could be fine when its cold but a bit of heat could cause it to move so its not making a connection.
I'd start with a cool board and start pressing on and wiggling chips
very carefully as to not short any thing out but to also see if you can cause a game to freeze because of a loose connection
sadly a board swap isn't possible with out a lot of modification to a toaster board, the whole right side of the board is different, that blue connector is just 5 connections. if I remember right that would be power button, reset and LED.
the sharp NES board has 9 connections where the blue one is at on the toaster and it looks like they re routed video and power input to come through the same spot and got rid of the connections in the bottom corner for the power and video thingy box.