I'd vote that the lift pump is weak and the higher RPM from a pull start brings just enough fuel to the injection pump. As a test that might work, set the injection pump to the run position, run the manual prime then crank.

For a long term fix, I'd use a inline electric pump tied to a 2 terminal oil pressure switch that shuts off when oil pressure is zero and a cranking bypass so it runs while cranking. Even better is tying the bypass to the glow circuit or just a manual prime button.

I put electric pumps on lots of the industrial / construction equipment I work on and found it eliminates a long crank on worn injection systems.

Depending on engine / line size, here is what I use. These are around $ 30 from rock a


AIRTEX E8012S Electric Pump; Solenoid; 12 volts; 5-9 PSI; 30 GHP; 5/16 hose; Domestic Carbureted, 5/16 hose

AIRTEX E8090 "Solenoid; 12 volts; 5-9 PSI; 30 GHP; 3/8 hose; Domestic Carbureted, 3/8 hose"

AIRTEX E8251 Solenoid; 12 volts; 2.5-4.5 PSI; 30 GHP; 5/16 hose; Marine Low Pressure