Some Williams coils like yours don't come fitted with diodes. They are a must in these Stern games. The missing coil diode is the reason the transistor keeps blowing. The banded side of the diode goes on the yellow wired coil lug.

Is the right coil being driven by a different driver (model) transistor?

When the driver transistor switches off and disconnects power from the coil, the coil has stored energy which comes out as very high voltage in reverse direction whose path goes across the driver transistor. This high voltage is beyond the spec of the transistor so it blows. The purpose of the diode on the coil is to clamp this high reverse voltage to a negligible level. The phenomenon is called "back EMF".

In this video, you can see when the user opens the switch, the back EMF from the large coil releases into the lamp causing it to briefly illuminate very brightly.