If you are able to start the engine everytime and after it's running it then shuts down, it doesn't seem to indicate a problem with the ignition switch. Though recalcitrant, it works.
If the fuel supply is cutting out after the switch returns from START to ON, then I'm with @sjoost. The problem sounds like it's related to a condition needing to be satisfied after start - the condition being a matter of safety.
If you an ammeter, look at the current to the pump compared to time. When is it on, when is it off. If it's repeatable, then it is under control, and suggests something is telling the ECU to shut it down.