As suggested above, it may be that it just takes too many pulls to move the gas to the carburetor. Also, I think that if good gas doesn’t start flowing quickly, it soon becomes diluted making it more difficult to get the proper gas/fuel mixture. Furthermore, gas is only being produced when there is suction on the system. That hesitation just complicates things even more.
I sometimes had trouble starting my rope-start push mower on charcoal until I moved my suction blower so it would pull the gas right over top of where the air originally entered the carburetor. After starting the engine, I remove the blower hose and screw on a filter in place of it. The point is to get the gas to the carburetor, before pulling the rope, either by pushing or pulling the gas there. Early on I was impressed by a YouTube video of a guy nicknamed “One-Pull Al.” After that modification I became “One-Pull Steve.”
In my case, I have a T-fitting over the carburetor. The stem of the T fits where the air filter originally was. The gas line fits on one side of the T, the air mixture valve and then the suction blower are on the other side. I open the air valve all the way and run the blower until I can get a flare. Then I remove the blower hose, adjust the mixture valve to a predetermined sweet spot, pull the rope, and away we go. You still have to determine the sweet spot, of course, but I think it is easier this way because good gas is already at the carburetor when I pull the rope. It works so well that I set up my Toyota to start the same way. I’m not sure of your configuration, but something working on similar principles might work for you as well.