It can be frustrating to figure out, but it is a simple system - the pump runs and injects unfiltered air into the exhaust manifold to provide more O2 so the cats will heat faster. The pump only runs at startup and then only when the ECU wants it to.

If you watch the O2 sensors with an OBD reader that displays live data, you can see the change when the pump runs - now, the pump only runs on startup and then only when the ECU wants it to. But you can open the RCM up and manually close the relay with your finger while someone watches the O2 sensor readings - if the O2 sensors show no change, the cause is:

1) The pump is not running (due to bad pump motor, etc. - easy to fix)
2) The pump is running but one or more of the check valves is not allowing air to flow or the pump itself is bad. (Changing the check valves would be, for me, challenging to do...)

The dealer will not be much help and the DRB would be OK, but if you have live data displayed in your OBD reader, that does the job.