It sounds, from the title of your thread, that you have identified the problem. If there is not enough fuel in the carb bowl when you start the engine, and it does not flow into the bowl fast enough, your description of the problem would fit. You have already changed most of the things that can affect fuel flow. Have you replaced the in-line fuel filter? Have you checked to see that the fuel shut-off valve is clear and open? When the engine dies from lack of fuel, shut off the fuel from the tank and see if there is a full carb bowl of fuel or if it is empty (I know, that means taking the carburetor apart or removing the bowl -- PIA). If no fuel then the problem is still in the supply lines or the float assy (I assume you replaced the carb because of the problem? not for some other reason?). Last thought;
does this particular carb have an anti backfire solenoid on the bottom of the float bowl? If that is not working, that could cause no fuel from carb bowl to engine. It closes off the main jet.