Amazing. After a long and frustrating period, my BlackBerry has been restored to its former fabulousness. I had lots of help from the team at RIM, but ultimately the only solution was a total and complete reset. We reflashed the BB with the latest software, removed my BlackBerry Enterprise Server account, and rebuilt everything from the ground up.
So what happened? I'm guessing that one of the many pieces of software I've installed and removed on this device resulted in some kind of corruption on the device. BES, being a good corporate tool, backs up each berry's configuration and data. The several attempts I made to resolve the problem by flashing the device only failed, because as soon as it was connected up to the BES again, my old corrupted files were loaded back onto the device. The only solution was a complete restart.
Many thanks to Jim Courtney, and the folks at RIM who helped me with this. Here's one little geek who's learned to take it easy on loading oddball software onto his BlackBerry.