I've got a similar problem with my elderly (at least 20 years) Yard-Man mower w/ 3.75 Briggs engine and Walbro carburetor.
Toward the end of this mowing season, it wouldn't start, period. So I:
--cleaned spark plug and air filter
--removed carb and confirmed that gas was flowing from the tank to the carb
--took apart carb and there was plenty of fuel in the bowl, so the needle valve wasn't stuck.
--the carb itself was reasonably clean, but a couple of small holes were clogged. Unclogged everything, cleaned w/ carb cleaner and re-assembled. The o-ring between the carb and the engine manifold was in good shape.
--put everything back together, gave the priming bulb a couple of pumps, and the engine fired right up on the first pull
--ran fine for about five minutes, then started to "hunt" mildly, then gently died. More severe hunting has been noticed for several years, but not to the point of the engine dying.
--if I push the priming bulb a couple of times, it will start immediately and run from 2 to 5 minutes before gently dying again -- sometimes after mild hunting, sometimes after running normally. This process can be repeated indefinitely, as far as I can tell.
I admit that one thing I haven't tried (yet) is to check whether the gas cap is plugged.
Also, I've never fooled with any of the adjusting screws (2) on the carb. And the throttle cable gave up the ghost several years ago, so I wired the linkage so the engine runs at approx 80 percent of full throttle, which works fine.
Any thoughts?