It turns out that the documentation for `radio' types explicitly
mentions what we probably want here:

,----
| ‘(radio ELEMENT-TYPES...)’
|      This is similar to ‘choice’, except that the choices are displayed
|      using radio buttons rather than a menu.  This has the advantage of
|      displaying documentation for the choices when applicable and so is
|      often a good choice for a choice between constant functions
|      (‘function-item’ customization types).
`----

So it looks like we want to change the definition to: