I decided to try using the free Motorola Media Link software and see if that worked. I deleted the Music directory on the SD card and used MML to sync some tunes from my hard drive to the phone. It worked fine; all the tunes show up on the phone. Interestingly, they're in the Music folder on the SD card, right where I had tried manually copying files.
So I don't understand how Music Player decides when to refresh its lists, but I have a working solution.
If you copy the files into the music directory and use SDrescan, you don't need to use the Media Link. Media Link would not run on my 64bit Windows 7 PC.