Lawn mower won't start/stay running?

Maybe the fuel is not reaching the carburetor, that is sometimes caused by a plugged vent in the fuel cap. Remove the line from the carburetor and check the flow. Then I would check for proper compression. Another thing is to clean the blade and make sure it was sharp. When you removed carburetor the gasket should also be replaced. Briggs has a Service and Maintenance manual that will give specifications for their engines. Try Amazon for one. The loss of power may be caused by timing or governor problems. the governor linkage may need oil to move free, the timing needs the manual and a timing light.

With the additional information about the priming it is certain that the carburetor is the problem. I would now suggest a replacement unit probably less than $50. There is a possibility that the primer bulb is leaking and should be replaced. Any gasket that is in the intake system will give problems.

What you describe sounds like a dirty carburetor. That is the most common issue with small engines. The carb needs to be removed and taken apart then soaked in a quality parts cleaner. Sometimes you can get by with a spray can of carb cleaner. The ignition timing doesn't sound like the issue to me. Timing is controlled by a key between the flywheel and the crankshaft. Hitting rocks etc can shear that key. But the starting and running for a few minutes doesn't sound like the symptoms of a sheared key. I think you need to do a better cleaning of the carburetor.

If you're too stupid to repair it, why are you also so stupid that you won't bring it to a repair shop? Those guys are much smarter than you, moron.