The solution to this was cryptic. Perhaps typical of real puzzlers, it was interference from completely unrelated software.
I noticed my Palm would not synchronize, which it usually does by Bluetooth. I checked which port the PC end was set to use. It was COM27!
I remember this now. If HotSync Manager ("HSM", the software that the Palm system uses to synchronize between a PC and mobile devices) is active, when you create a new COM port, HSM assumes you want to start using that port for any local synchronization.
This is the first time I have been mad at HSM, but it was a real brute in this case. I could manually change HSM back to the Bluetooth COM port it was supposed to use, but every time I would plug in the mbed, HSM would grab that port again. The solution is to exit HSM entirely whenever I am working with the mbed.
I uninstalled COM27, ran mbedWinSerial_16466.exe with HSM off. This time it installed COM3 just fine. The serial test program is running as it is supposed to.
This might also be a problem if you are using other automatic synchronization software. I believe the Microsoft analog is called something like "ActiveSync".
I ran the installer (mbedWinSerial_16466.exe) (on Windows XP Pro, Version 2002, Service Pack 3). A new port (mbed Serial Port, COM3) appeared, but with the error "This device cannot start. (Code 10)".
In Device Manager, the port disappears/reappears when I unplug/replug the mbed; but always shows the yellow-circle-and-exclamation-point icon indicating a problem.
Hyperterminal will let me configure a session using this port and even acts like it connects to the port, but no characters ever come in from my test program.
I have tried uninstalling and reinstalling the driver several times, using different cables and on different USB ports. I have tried re-downloading the installer using Chrome, FireFox, and IE, including direct Run rather than Save (IE). I have made sure there was no Explorer window open looking at mbed. Always the same result.
I researched "Code 10". It means "there was some problem on a deep nether level".
I am out of ideas on how to investigate this further.