I got an e-mail from Wolfram Support, and suggested that I un-install Java. As I discovered when doing this, The Java 8 installer (for the Mac, at least) just installs a plug-in. It does not install the JDK, as, apparently, it used to.
The strange thing is that once I uninstalled Java 8, Mathematica 10.1 still evaluated EntityValue correctly. As the tech support person pointed out, how Mathematica works should not be influenced by what is essentially a browser plug-in.
Kenneth's observation is correct: when the function was not working, the blue box indicating that the WolframKnowledgebase connection was being initialized did not appear. Now, it does.
So, what I am left with is a working Mathematica 10.1 after some voodoo that was a side effect of installing Java 8 on OS X. Now that it works, the presence or absence of the plug-in makes no difference.
Completely resetting Mathematica 10.1 (deleting the base directories, re-installing from a fresh download) did not fix the problem. Whatever it did, running the Java 8 installer made the problem go away.
Since others have tried this bit of magic without success, clearly something else is going on, to state the obvious. If, as Marco discovered, CityData works and EntityValue does not, accessing the same data, the issue cannot be in the connection to the knowledge base.
I'm out of ideas. I had a magic trick that worked for me (good for me) but no idea why, or what to do to solve the problem.