I backed up my 4S to my laptop, and went to restore it. I was surprised when the apps didn't restore, but apparently you have to select the option to Transfer Purchases for apps to backup and restore.
In any case, I eventually backed up my apps from the old phone, and then restored them to the new phone. Unfortunately, I had already downloaded and installed some apps, so all the apps were installed to the new phone, but no icon placement restored. So, I spent an hour or so restoring my icons. Then, I went to work trying to log back into the various apps, transfer settings, etc.
One of the games I played didn't allow for logins, but instead relies upon the device serial. The account transfer procedure for this was to download another one of the company's games, which had logins, which would update the setting for all the related games. I'm guessing this would set some local database value on my phone and update their servers with my device serial. So, I went to download the new game, clicking on the install button in the App Store. I watched the progress icon rotate and the app install, then clicked on the "open" button, but nothing happened. I tried clicking this button a few more times, and nothing happened, so I ended up powering down and powering up.
When I powered the phone back on, most of the app icons that I had spent the previous hour setting their locations were gone. I was confused, so I went to the App Store again, trying to reinstall one of the apps that was no longer there. The App Store was showing the "Open" button for the apps, implying that they were already installed. I then noticed this in my email:
"Your Apple ID, [redacted], was just used to download Racing Live™ from the App Store on a computer or device that had not previously been associated with that Apple ID."
So, something had obviously gone wrong. Next, I went and checked the Settings > General> Usage, and the picture is what I saw. When I clicked on one of the apps without a name, the name showed as "(null)". I don't know much about the backend for iOS, objective C, or the databases that are used, but apparently something had gotten corrupted.
Blah. I don't know what I did that triggered this bug, but it wasn't good. The only way for me to recover was to reset the 5S to defaults and re-restore the original 4S backup. The one advantage to this process was that since the apps were backed up to iTunes, they all restored . . . but far from the seamless upgrade process Apple talks about . . I'm going to guess I probably spent 8 hours or so dealing with the various issues of backing up, restoring, backing up the apps, restoring the apps, organizing the icons, researching if there was a way to repair the app database, and then having to do it all over again . . . blah.