I am not certain, but on later Copals the self timer latch is at the upper end of the unit as shown in your photo. At about 11 o'clock on your top photo is a screw on the body that is just past the self timer unit itself. On your photo withj the spring going straight left, you can see a latch connected to the pallet. Most likely this latch is pushed in a way that hold the pallet against the star gear and stops the self timer from running down.
All of this happens if the shutter is already cocked. Shutter not cocked= no self timer action (or possibly simple run-down on an older shutter like this?)
So anyway, focus on the pallet and its latch, and how this intereacts with the shutter cocking and release mechanism.