I fixed a Honda tiller with similar behavior. Ran fine for a couple minutes then stalled out. The gas cap was repeatedly put on so tight that the gasket was permanently smooshed out so much it sealed the cap up completely. On most lawn equipment, the cap is designed to prevent gas from sloshing out or leaking out if you overturn the motor but it is also a vent that must allow air in. On the tiller, the cap would not allow air in to the tank to replace the volume of gasoline being used. After running a few minutes, the tank was actually under vacuum which prevents gas from going out.The fix is a new cap, cap gasket or cut some small nicks in the edge of the gasket to let air in. Cheap fix or at least a cheap try to fix. Since you have a new cap, try not putting it on so tight or cutting some small v shaped nicks in the cap gasket. Perhaps the new cap was the wrong model. You might try loosening the cap and start it up to see how it runs. If it seems okay for a couple minutes of idling, then tighten it and see what happens.
A partially blocked fuel line or tank exit might allow it to fill the carb bowl, start up, but might not allow enough flow to keep running. Some claim that alcohol in the gas can cause the various rubber/plastic formulas used to swell or rot or gum up. I've seen rubber fuel lines that were soft and mushy, dried hard and cracked, or permanently kinked. A new fuel line is another simple and low cost item to replace.
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