That lamp has to be in the circuit or the alternator won't energize. Sometimes if you rev it up there is enough residual magnetism to get it going. Then it will charge till you shut it off. But generally the reason the red light comes on with the ignition is to tell the driver that the bulb is good. If the bulb is good that alternator gets a mild 12v signal to energize the field coils. Better get that dash light fixed.
This sounds promising.