There has been a discussion on this topic not so long ago.
As we have seen, the system works differently here than, for example, in word processing programs such as LibreOffice, where templates can also have content for newly created documents. I think that this experience has led to some misunderstandings here.
A look at the file structure of Aeon Timeline shows that, among other things, there is a part for the project setup that defines the possible item types with their allowed relationships, and another part that describes the concrete instances as they are created by the user and arranged in the diagrams.
A template is, if I don’t see it wrong, nothing more than a file that contains all the information for the setup part. If you wanted to add diagrams for exemplary structures, you would technically already have a full project.
As a template combining setup and content, as you have in mind, a project file would indeed come into question, as you mentioned at the end of your post. To make it easier to use, there could be a menu item like “Create file from sample project”.
Edit:
The template also contains the definition of the calendar system. This, however, can be seen independently of narrative patterns.