Ok, I'm leaning towards the piston scoring being the problem and nothing else. Turn off the fuel, remove the plug and pull the rope 20 times or so to clear out the excess fuel in the cylinder and crankcase. Replace the plug with a known working plug or a new one and try to start the mower ( still with the fuel off). If you get no love, give it a shot of starter fluid, and try to restart. If you get nothing and all that you tell us is correct then the engine is too far scored and not creating enough positive and negative crankcase pressure to start and most likely never will unless you rebuild it.
If it does start then turn on the fuel and let it run to see what it does and you can further diagnose any carby issues, etc thereafter.
There is no need to rebuild or modify the carby until you've seen if it shows any signs of life by using the above mentioned technique.
