here is one idea:
check the shut off switch that is controlled by the bale on the handlebar. Due to wear, weak linkage, or corroded cable, the contact may still be grounding the ignition.

2nd idea:
remove the air filter and examine the venturi tube while depressing the primer bulb. You should see fuel pumped from the float bowl. Sometimes, the gasket between the carburetor and air filter housing deteriorates and the primer bulb is not effective.

When you are having the problem, check for spark.
IF you have spark, prime the carb with a couple teaspoons of gas and see if it fires.
That way you can isolate the problem to fuel or electrical.

Providing the engine Model & Type numbers would allow one to know WHICH Briggs you have.
Knowing which type of carb used simplifies things greatly.