After spending almost an hour trying to understand why an extended method would just not be available to the extended object types, I found out about a limit of this feature that I will make a note of, for future reference.
Compared to Java and C/C++, the switch statement as implemented in the C# language offers more flexibility. It can be used with a wider range of types and offers a more complex fall-through mechanism.
Those of you who use WordPress and installed it themselves will probably know that the title for this post is the default one you get after setting the software up. But I had a good reason for keeping it, laziness aside.