An inductive pickup works on current, not voltage... once the arc at the plug is initiated, the gap resistance drops to a very low value. (Swamped by the resistor plug or resistor wire if used). Then the energy in the coil is delivered over time. (Yes, there's more to it but this is simplified as much as possible)
Most likely the problem is (as the OP noted) the clamp probably isn't closing all the way. Even a very small air gap of thousandths in a transformer core causes most of the flux to appear across the gap, so the secondary (the pickup winding) will not see much signal to trigger the light. Make sure the clamp is completely closed and that the core ends are free of debris