heres a guess at why this happens, i can remember a few of these same situations the carb is not letting fuel pass. something to do with carb make up. the weak tabs on carb parts are pushed closed by the heat coming from the case thru pulse channel, which normally pulls the fuels.
its a limitation of carb (stock). the hi performance carbs don't seem to have same issues.
thats a guess, truly think its a stock carb problem where low duty parts placed in hi stress duty situation just can't keep stable positions , small tabs, thin diaphrams, pins closures under stress won't work right...as the motor is running the fuel is "COOLING" all these parts , when its stops temp will spike and pinch close.
compression falling off would be the next concern. if compression is weak and runs fine first couple mins , then starts to fade a bit , thats definite compression problem. and with heat and part growth it all grows apart and there is no compression.