An engine needs three things to operate properly: correct air/fuel ratio, sufficient spark at the right time and compression. It sounds like one of the above is intermittently missing from the picture and intermittent problems are the most difficult to find.
If I am understanding your original post correctly, it sounds like the mower will work just fine for 30 minutes and then die. When you attempt to restart the engine it will run for a moment and die again with this process repeating itself until you let it cool down for 30 minutes and then it will run just fine for another 30 minutes. Is this correct? If so, I would check for spark while the problem is evident. It is possible that the ignition coil is heating up and going open while hot and when it cools down it will work again. A bad coil is not the most common problem but it does happen from time to time.
Since the engine runs well for a period of 30 minutes and then dies, it leads me to believe that it is spark or fuel related. However, a sticking valve could cause a loss of compression and then free up after it cools down causing the symptom you describe. As I already stated, one of the three factors is missing and you are the eyes & hands of the situation which means you will have to do the work to find out what is missing.
Make the problem happen and check for spark first as it's the simplest system to check. If the spark is good then spray a shot of carburetor cleaner directly into the carburetor (air filter removed) and try to start it using carburetor cleaner as the fuel.