How does spiral cable assembly relate to airbag

The passenger air bag assembly would be the inflator and bag with the wire kit being the short harness that connects it to the SRS system.
The spiral cable aka clock-spring, is behind the steering wheel. It has wires for the drivers airbag, horn and any steering wheel controls in it.
Both are part of the Supplemental Restraint System aka airbags. You also have side curtain bags and explosive pre-tensioners in the seat belts.
If any part of the system fails the tests that are done when the key is turned on or the continuous tests that are done when the engine is running the entire system becomes disabled. In your case the cable failed, it is somewhat common as it has to flex every time you turn the steering wheel. The spiral cable would not be replaced until it showed a fault. It is not something that was covered under the airbag recall either as it was not the failure item.