The ECU does control the solenoid. Anything that controls the solenoid can be an issue, if it is really a heat related issue- could point back to the ECU. Does the time extend itself with the hood off? It too is fairly common for the pick up tube in the fuel tank to get partially plugged with grass and debris which causes an intermittent stall when the pump can't overcome the debris. Pine needles and strands of grass are the worse for this. Cleaning the fuse block and switch terminals with CRC should be done anyways. I'm not sure if you are servicing mowers for living, because if you do- you should have an ECU in stock.