I would try using the "broken NES" with the 72 pin connector from the working NES, if it works, thats the issue. if it doesn't, then I would try replacing the mother board or try plugging in the cartridges in to it while disassembled to see if the cartridge loading mechanism doesn't make a good connection.
Have you tried cleaning the sim card/tray and or blowing on it? It might sound silly but the water vapor in your breath can strengthen electrical connections between the sim card and tray.