How to add an autocomplete field to a form
Well, that was pretty ugly, and a LOT of guesswork is going on here.  But:

When Drupal sets up an autocomplete field, it adds a hidden variable to the form, which (I'm guessing here) is used to track the status of the whole autocomplete thing.  If you inspect a rendered form, it will look something like:

&lt;input type="hidden" id="edit-field-name-autocomplete" 
      value="https://example.com/path/to/autocomplete/handler" disabled="disabled" 
      class="autocomplete autocomplete-processed"&gt;


That autocomplete-processed field seems to be the key: I had blindly copied it into my code and made it part of the field I was added to the form.  Once I removed it -- autocomplete-processed, that is -- autocompletion ran properly on it and all other similar fields.

Whew.  Your mileage may DEFINITELY vary.