Thank you for the clarification.
Just to explain the behavior correctly — the module is not designed to list all the places directly. It always shows only the country first. Since the module is structured to display the country layer only once, it will not list all the places at the same time. After clicking the country, it then loads and shows all available places under that country.
So the issue isn’t with your data — it’s simply how the module is intended to work. It shows the country once, and the detailed places appear only after the country is selected.