My understanding of 1.6 engine controls:
1. ECU requires air flow input from AFM.
2. ECU requires air temp input from AFM.
3. ECU requires WOT switch input from throttle body.
4. ECU requires crank angle input from the CAS in the back of the cylinder head.
If ECU does not sense these inputs within normal operating ranges within ~3 seconds of initial start, then it will kill the ignition pulses to the coil. The ECU will then log this failure in internal volatile memory. The ECU will refuse to send ignition pulses while this failure is logged. I am not clear if there is a timeout on this failure, as I haven't tested it rigorously enough to be able to tell. Removing unswitched power from the ECU clears this logged failure---at which point the ECU will again attempt to start the engine, and the cycle repeats.
AFM has a reed switch that is in series with the fuel-pump ground. If the flapper door is fully closed the reed switch will prevent fuel pump from flowing. If you take the cover off the AFM there is a little metal arm that opens/closes the reed switch, this can be bent out of the way to prevent it from ever shutting off the FP. NOTE: Losing fuel PUMP will not result in an instantaneous engine shut-off. Only once fuel pressure bleeds will engine sputter and die. I went through this exact failure mode at Hallett back in June. At idle, it takes quite a long time to bleed pressure away. Even at WOT, the engine leans out and sags in power...but, doesn't "die" instantly.
The fuel injectors are powered by switched power directly from the main relay. The ECU injector "pulse" grounds each injector to energize the solenoids in pairs (1,3) or (2, 4). There is nothing else involved electrically in the 1.6 injectors (different from the 1.8 described by Ron above).
The ignition coils receive ignition pulses from the igniter, similar to the fuel injectors, these are also in pairs (1,3) and (2, 4). These pulses are again GROUNDing of the coil. The coils are energized by switched power from the ignition switch. There is a capacitor across the coil, which is critical for proper ignition operation. The igniter receives pulses directly from the ECU.
If you need NA scematics, you can find them in the downloads section of my "build" link below.
I stand by my theory above, that it is likely a sensor input to the ECU failure (broken wire, failed sensor, etc). To (dis)prove this, disconnect the battery for a few seconds, and then reconnect and start the car. If you get a short "run time", followed by none...every time you disconnect the battery, then you have a bad sensor input, 1-4 listed above).
If its not the ECU shutting off the ignition, then my next most likely culprit would be the igniter....its really the only other component that is COMMON to all 4 cylinders (or really the two cylinder pairs).
After that, it could be the condenser capacitor that is across the coil power. A bad capacitor could initially hold a charge, but then progressively fail to maintain that charge eventually resulting in a loss of spark. This COULD manifest itself in a semi-random run-time depending on the time between attempts, and any progressive failure mode of the capacitor materials.