Another follow-up, and perhaps final thoughts on this issue:
Into the next day now with a new phone and clean install, setup as-new, not from a backup and the Touch ID is working as designed. That is to say, I'm getting about 90% or better on all prints. I have not seen any degradation since yesterday. If anyone is still having problems, I suggest a clean install of the software using iTunes (not OTA) and setting up the phone as-new. If that still doesn't work, call Apple or visit an Apple Store and have them swap out the phone. There's either a hardware issue or corrupted firmware that can't be fixed by a restore (or some other issue that Apple has yet to solve).
When you set up the prints, make sure the sensor is clean and dry and your hands are relatively clean and in the state they will be in most of the day...not too wet or dry. Follow the directions here to setup the prints properly: http://support.apple.com/kb/HT5883. The biggest source of failed recognition seems to be wetness (like after a shower or excercise) or very dry skin. After a shower or bath, it make take quite a while for your prints to get back to their regular state where Touch ID will again work reliably.
Incidentally, in my earlier post about calling Apple Care, I wrote that when the tech ran a remote diagnostic it showed the phone running iOS version 7.0.1, even though it was at 7.0.2. This was a red herring. The Apple Genius at my local store said their diagnostic tool was making that error on all phones, so it had nothing to do with any corruption in my particular setup.
In the end, this technology is new to iPhone, and as with anything, it will improve with future software and hardware updates. It works pretty well, but it's not perfect. However, it is better than anything else out there because at this time no other phone even has fingerprint recognition. It makes the phone a lot more secure for those that previously did not use any passcode lock at all, which apparently was more than half of iPhone users.
I will post again if I have any further trouble, but I think my issue is about as solved as it can be. Thanks to all.