I’m not there, I can’t hear the engine run, so I must consider all possibilities. I guess my next test would be to pump the bulb, then just let the engine run, time how long it runs before quitting without helping it along by priming the primer bulb. Next, I would pump the bulb up, then start the engine and see how long you can keep it running by helping it with the primer bulb. Again, I’m not there, can’t hear the engine. You should be able to keep the engine running pretty much continuously by priming the bulb if the fuel pump is struggling or there is an issue with the fuel supply system (there are exceptions of course, but generally speaking).
Did you replace that relatively new fuel pump with the smaller one in your latest picture? That little pump does not look oriented correctly either, there should be a little plastic tab on the housing indicating “up”. Having the original filter/bowl is a “nice to have”, but not really necessary. There is a little screen in the fuel pump cover, but the external filter/bowl set is nice to show water or other crud in the liquid fuel, easy to spot in the bowl.
The nozzle gasket has been controversial on these style carbs for a long time. The older models did not use nozzle gaskets, not sure if yours came equipped with one from the factory though. Using a nozzle gasket, provided it fits properly, seems like a good idea, really don’t think it could hurt anything, only help. Nonetheless, fuel flow up the nozzle doesn’t begin until mid range throttle, the idle passages are the small cast passages in the bowl and up through the side of the upper body. So, in most cases, I don’t think the lack of a nozzle gasket is going to affect running at low speeds, even if the engine is supposed to have a nozzle gasket. Once the RPM has been increased a bit, fuel begins to flow/get drawn up the main nozzle, it is then when the extra fuel (no nozzle gasket) could flood the engine/cause rich running. If you pull that carb apart again, please post pictures with the bowl removed and the float installed.
All that being said, it is surely possible that the engine is losing spark on one/both cylinders after running for a few minutes, causing it to stall out. I like Bill’s idea about disconnecting the vacuum switch, although I’m not too sure just which lead needs to be removed/isolated, would have to look at a wiring diagram. There could be a slight short in the stop circuit as well causing loss of ignition. I guess the easiest was to confirm this would to be disconnecting both leads coming down and out of the mag plate, be sure to tape them up though, don’t let them touch ground. The engine is running slowly enough while you are testing it that loss of spark to one cylinder would probably cause the engine to stall. So, does the engine just “shut off”, like someone turned the key off? Does it seem to misfire before shutting off (ignition issue), or does is sneeze/cough then stall out?
I guess I would try to get the engine on a boat or in a decent tank where you could get it in forward gear and advance the throttle to see if it will remain running at higher RPM. Ignition/fuel flow issues are easier to hear when the engine is running at high RPM. You will hear it if there is an ignition issue and it drops a cylinder. Lack of fuel will be noted by surging/bogging.
Again, we have to be careful to stick to the basics, using concrete testing procedures, and doing our best to avoid inaccurate conclusions based on assumptions and improper/incomplete testing procedures.
