I'd try disconnecting the ground wire that comes out from under the flywheel cover, and try it then. It will be the wire without the bulge in it. (The bulge is in the charging diode wire)
And, there could be dirt collected in the gas tank, and flowing into the fuel line and carb. Or dirt in the carb jets, Or a disintegrating fuel supply hose. Or a bug egg in the gas tank cap breather hole. Check the seat switch, by sticking a piece of cardboard, or a stick in it to keep it open. Did you clean the legs of the coil, and the flywheel magnets. Take it off, sand the coil legs, and the magnets on the flywheel, then reset the coil leg gap, using a business card to get the correct gap of the coil to flywheel setting. (Might sound "hokey" but thats how it is done by us semi-experts! I've tried the thin guages that mechanics use, but they are frustrating to use.)
Check that the clutch safety switch hasn't fallen away from where it is fastened on.
On one of those Briggs, which foiled all of my attempts at making it run, i installed a switch in the wire i mentioned above, to shut down the engine. Also, if it has the fuel shut-off thing on the carb bowl, make sure it clicks. It might be stuck with gummy gunk from sitting. It can be removed and cleaned. Best to check it, and fix it if possible--those switches are plumb expensive! $80 was one price i have heard! I'd think that might be your problem, as gasoline will tend to gum up, and make that switch stick, only allowing fuel to seep into the carb bowl, instead of flowing freely!
Got to go! Going Bluegrassin' this afternoon!
Rustyj

I too changed plastic flywheel to aluminum. It ran ok untell gas mixed in with oil. put new carburetor on, now its hard to start wont stay running, rpm with more gas does not rev up. Engine dies, backfires, exhust system gets hot and smokes. won;t idel with dying. Any suggestions need help please.
Joe