Sounds like you are on the right track on diagnosing the problem. Several other things to check: First, loosen the fuel cap to make sure it's airway is not stopped up.Another thing is to have an assistant to turn the ignition switch on while you are standing close to the carb side of the engine - you should hear a sharp click one time coming from the bottom of the carb. If not, the afterfire solenoid is gummed up and sticking. Repair.
Next, drain the fuel tank of fuel and replace the flexible fuel line from the tank to the fuel pump (the old line may be getting soft and collapsing). disconnect fuel line at carb, crank engine to verify good fuel stream being pumped to carb. If not, retest to determine if fuel pump or fuel filter faulty (replace either with only OEM parts!)and replace balance of flexible fuel line.
Once good fuel flow established to carb, engine should start/run. If not, then time to remove/clean carb and it's afterfire solenoid located at carb bottom. Get a carb rebuild kit, a magnifier glass, some stiff brass 20 ga wire and carb cleaner. take pleny of before/after pixs to know how to get all back together, take carb fully apart, soak parts in cleaner overnite, use magnifier and brass wire to probe all passageways in carb parts, compressed air to blow out parts; rebuild with fresh parts from kit as instructed. Patience precision,persistence.
remount carb back to engine, remake all connections. Test for for proper start.