Been reading this thread and thinking about your problem and hope this helps you figure it out.

Take a breath and go to back to basics. If you have good compression, spark, and fuel it should run. I think you have compression because you got it running. I think you have spark also since it ran I think your problem is fuel delivery.

If you have consistent good blue spark on both spark plugs when you pull the starter rope then I wouldn’t think you don’t have a problem with the ignition. You said if you push the primer over and over the motor will continue to run. Is that the case or maybe not? If you can make it keep running by pumping the fuel primer button then your problem is with the pressure system of fuel tank. Someone here also mentioned to make sure you got your fuel line and pressure line not mixed up when you pump primer button is should be filling the carb bowl not the engine

Also the engine is starting and speeding up which means you must have decent spark and as the motor runs it is starving for fuel that speeds it up until it stops running.

You said the motor speeds up on its own when you do get it started that suggests to me it is starving for fuel not enough in the carb bowl or its not pumping it into the carb bowl?? Carb float maybe not adjusted correctly or needle jet/fuel entrance into carb is clogged not letting fuel into the carb . Also you could have a clogged high or low jet in carb. Have you tried pumping fuel into the carb using the primer button then carefully take carb off over a plastic container and see just how much fuel was in the actual bowl.

I would try cleaning the carb next check the float adjustments, make sure when you pump primer fuel is going into the bowl and it is actually enough to fill the bowl. Did you put a carb kit in the motor?? When I work on the carbs that have part number 0344037 I have noticed that particular part has to be in great shape or the motor can not draw enough fuel into the engine to run correctly especially at high speeds.

Hope that helps,
Joe