So ya gotta do the "rock-a-doodle" with it eh...LOL
I would think this would point to the carb. It sounds like the vent is plugged???? Suck on the vent line. ...Suck twice~spit once..LMAO!!
If you leave the petcock on prime is it OK (vaccum working?)??

Once started and run, there is no fuel residue in the motor.

Instead of rocking it, try hand choking the carb with 2 fingers and see what it does.