The mechanical fuel pump has built in check valves that are not only a must for it to pump period,but to hold the fuel from draining back into the tank--they could be faulty and working well enough to pump fuel,but not hold it with the engine off.
Also if the diaphram in the pump has a pinhole or rupture it can still pump,but also let gas drain into the crankcase when the engine is off,and that can do nasty things to bearings & rings due to the oil being diluted..
Even if the fuel did drain back,there should be plenty left in the float bowl to start and run the engine at least a minute also--perhaps the carb needs an insulating spacer to prevent heat soaking into the carb and boiling off the gas--ethanol gas evaporates rapidly when it gets too hot,much faster than the old leaded gas these engines were designed to run on..