The flaps are the one-way valves. In one direction, fuel pushes against it and it opens; in the other direction, fuel pushes against it but the flap hits the metal so it doesn't open. That's how it's supposed to work. What's probably happening for you is that a sip of fuel is sucked up the long tube, but then is pushed back down the same tube, so the fuel never moves further over to the well.
So the answer is yes, that would cause the engine to stop. Get yourself a new diaphragm gasket.