Thank you, Boyd, for hanging in there with me on this. Of course, this is not how the eTrex is supposed to act. Garmin SELLS maps on MicroSD cards. The device is able to access them!
Just as I was ready to give up, I think I figured out what is going on. Allow me to explain the strange procedure:
I removed the MicroSD card from the device, placed it in a card reader plugged into my PC, and verified that it is formatted FAT32 and has no errors. No problem with the card.
As a last resort, I put five state maps on internal memory using BaseCamp, and then I added the British Contours map (by talkytoaster) to internal memory via drag-and-drop onto the drive. I felt as though I could settle for this arrangement if nothing else worked.
As one last try, I added an OpenStreetMap of the UK to the MicroSD card using drag-and-drop to the drive. Success! For the first time, the eTrex was able to access this map from MicroSD and list it in MapSetup>Select Map to enable or disable.
Out of curiosity, I looked into the drives and here is where it gets STRANGE:
The internal memory still contained five state maps, but these same maps got duplicated (automatically) onto the MicroSD card! I deleted them. Also, the OpenStreetMap got automatically moved from the MicroSD card to the eTrex internal memory! Thinking, "Oh, that is why the device could read it," I deleted it.
I could not stop troubleshooting at this point...
I put the OpenStreetMap back on the MicroSD card. This time, everything worked as it should: The eTrex accessed the card, none of the state maps got duplicated, and the OpenStreetMap didn't get moved to internal memory.
Finally, I opened BaseCamp and MapInstall to make sure the eTrex internal memory was represented correctly.
(I was afraid to breathe on the eTrex at this point.)
So, happy ending. Boyd, your original suggestion is good: Remove all maps (except the one that came with the unit) from internal memory and the MicroSD card. Start loading maps ONE AT A TIME. Now, KEEP CHECKING the drives to see where the maps go. Delete or move any maps that end up in the wrong places or are inaccessible. Keep segment limits in mind, and don't overload either the internal memory or the card. MapInstall is bossy about where the files go.
Why is this process so glitchy? My guess is that Garmin never intended anyone to make and install maps of their own.
I hope anyone who has problems with their eTrex will be helped by this thread.