After a couple hours on the phone with Apple Support Level 2, and then almost five hours in an Apple Store being helped by a succession of seven people, we finally just stumbled into a solution, and it is so easy, I'm actually unhappy we wasted all that time on this idiotic and unnecessary problem.
Here's what we did:
- Plug iPhone into USB port
- Open Control Panel
- Open Device Manager
- Expand Universal Serial Bus Controllers
- Right-click Apple Mobile Device USB Driver
- Choose Disable
- After driver is disabled, unplug phone
- Plug phone back in
- Right-click Apple Mobile Device USB Driver again
- Choose Enable
- Voilà! Your phone should now appear both in Windows Explorer and Device Manager (under Portable Devices)
I don't know whether you actually have to unplug the phone after disabling, but that's the step we took, so it won't hurt to leave that step in.
This has worked for both my iPhone and iPad, and now on two machines.
Good luck!