The mower shutsdown after the engine heats-up. You can mow for a short time and, just out of the blue, the engine will stop. No restarting at that point.

Well, are you able to restart the engine after it has cooled for a couple of hours? If so you have a coil (armature magneto) that is breaking down from the heat and will not work again until it is cooled. Get the mower started, mow, and when it dies, remove the spark plug, and check to see if the spark plug is getting fire from the coil. I believe you will not. If not, replace the coil.