Chip, First, the two relays you pictured should actually be the KS (Advance Unit relay) and the FS (00-90 Unit relay). AS just designates the generic type of relay Gottlieb used (Second, the KS has one of the filthiest copper circuit boards I've ever seen in a working game.
You pretty much understand how the bonus unit advance circuit is supposed to work.
So when you manually rotate the spinner with a game enabled, does the C relay (Spin Target relay) energize?
The C relay should have 2 normally open switches on it. The closure of one of those switches (one switch terminal has a red & white wire going to it) should result in the coil for the KS relay stepper getting energized.
If the C relay pulls in, but the coil of the KS relay NEVER energizes, then the switch on the C relay should be checked for adjustment and cleaning. Also check the switch tab ends to make sure the solder connections are intact and a wire has not broken off. If the KS relay coil still does not energize, the coil itself may be discontinuous.
If the C relay never pulls in, then the switch on the spinner is not functioning OR the wiring between the spinner and the C relay is not continuous.
I cannot recall: is the KS relay located in the backbox? If so, then the wire comes to it through a Jones plug. Check the wires on the Jones plugs and try reseating them.