Why Rod Core ignition coil in cars instead of E-I core or toroid core?
Think of it as a coupled inductors. The energy for the spark is stored in the coil during the period when the contact breaker (or transistor equivalent) is closed. When the breaker opens, the primary inductance and the capacitor across the breaker form a resonant circuit so the voltage rises to approximately 250 V in a roughly half sine waveform. This is transformed to some thousands of volts at the secondary for the spark. If the magnetic circuit were closed, the inductance would be higher, the rate of increase in current would be lower and the core would be more likely to saturate. In a capacitive discharge system the energy is stored in a capacitor rather than the coil so I imagine a transformer would work in this case.