Now Printing...

Well, being who I am, I have not yet RTFM. However, being who I am, I figured out the problem: HP's Installer and the Setup Assistant don't work over the network, only when directly connected to the printer in question.

What kind of sense does this make?  All the underlying software recognizes the fact that the printer is available -- the software "sees" it, sends a query and gets a reply -- but the installer refuses to interact with it unless the printer is directly connected to the computer via USB.

BUT! Once all the computers have been appropriately configured with the printer directly connected, you can then plug in the printer to the AirPort Extreme base station and (after RE-configuring as a Bonjour instead of USB connection) print just fine.

Oh, and the copier now works, too.