It's 4.5 hours later, and the Springboard icon is finally gone. All it took was:
(1) A complete restore of the phone.
(2) Setting the phone up as a new device.
(3) Manually selecting the apps I wanted to install (about 235 in all).
(4) Reorganizing the apps into folders using iTunes.
(5) Re-reorganizing the apps into folders, because iTunes just flat ignored the previous attempt (yes, I remembered to hit "apply").
(6) Relogging into all wifi accounts (this will take a few days to complete, as I move between home and various offices).
(7) Re-pairing with various BT devices
(8) Recreating all email accounts, contacts and calendar entries (thankfully, I use Exchange, or I would have been really hosed.)
(9) Relogging into all apps on the phone individually.
(10) Reautenticating with work, banking and other secure services.
(11) Blowing away all previous iCloud and iTunes backups for the phone.
(12) Creating new iCloud and iTunes backups.
All to get rid of one annoying little white icon.