Not due to inlet valve not seating, could be the opposite, inlet valve sticking closed. Pulling the engine over has to develop enough vacuum to pull the main diaphragm down to open the inlet valve, pull fuel up the fuel line, through the check valves in the pump diaphragm, through the inlet valve and fill the fuel chamber. Fuel pump action from crankcase pulses has no affect at cranking speeds, it's all about the choke which must be COMPLETELY closed. A main diaphragm that is too stiff to be pulled down enough, and the check valve tabs on the pump diaphragm being stiff and sticky can be a problem. This is assuming your starting procedure is good, the throttle has to be set to the partially open (fast idle) position and stay there until the engine runs. A cold saw will seldom start with a closed throttle, so it you blip the throttle after it pops, the throttle will be returned to idle and must be reset to fast idle.