The transistor is being fired from an IC (CA3081, a small one). That IC sometimes fails as well. But what I have experienced it almost never fails resulting in a coil being stuck on, but always the coil not firing anymore.

Next idea: is the slingshot connected to output 0 from the 74154? This is the 1-16 decoder, and it will fire output 0 if the inputs are 0000.

Try pulling off the coil-connectors, leaving J4 on the board. Then measure the signals (5V or 0V) on the inputs and outputs of the 1-16 decoder 74154. Put the machine in test, firing a coil, and compare with a working circuit.