On other games the leaf switch closed by the rotating weight (or spinning target) advances the mini stepper (AS relay) each time the leaf switch closes. In your game it sounds like the Spin Pulse relay could be another candidate. Either way you should be able to manually spin the weight slowly and see something else activate each time the leaf switch closes. If the leaf switch does drive the mini stepper it could be that the mini stepper (AS relay) is a little sluggish and not able to respond to the relatively quick pulses of the rotating weight.
On other games the delay relay allows the rotating weight to stop spinning before awarding whatever points or target the mini stepper advanced to so it might not be involved.
For anyone who can't visualize how the Spin Unit works there's more info and video here: www.funwithpinball.com/small-boards#SpinUnit
/Mark