For example, a birthDate where no "value" property is present, but instead there's an extension that conveys "refused to answer" or a string value for the date or the date in an alternate calendar system. Or an Identifier or Coding where only the display element is present but none of the components used in a 'standard' search. Are such elements expected to come back when someone searches the element using ":missing"?

My preference would be "yes" as that ensures that it's always possible to retrieve a complete set of data by querying on the different allowed values + :missing. Otherwise, a system that's trying to be complete and looks for male Patients with a name of John Smith and where the date of birth is either missing or falls within a specific range won't be foiled because the <birthDate> element was present but only had extensions.