I wouldn't personally replace the CDI without testing the other components (stator, pickup coil, electrical connections) first. Someone else more familiar with them would be better to tell you how to do this.
Most likely the spark plug wire is part of the ignition coil, so that will rule out everything past the CDI, except for the spark plug itself.
Some safety switches such as the kill switch might also allow it to turn without firing. Although it sounds like you tested this?