Thanks, Papa23! Fixed it!
1) Open the folder %USERPROFILE%\AppData\Roaming\Microsoft\Windows\SendTo.
2) Move all shortcuts out of there to another folder someplace memorable.
3) Do an Office repair (control panel, programs and features, click Office in list, click change button, choose repair, reboot at end).
4) Copy the shortcuts back from the memorable place to %USERPROFILE%\AppData\Roaming\Microsoft\Windows\SendTo.
5) Verify that SendTo works from Windows Explorer (opens an email message to send, as expected).