In test doc.odt case, you configured Default Page Style to offer different headers on first, left and right page. Then if you did not explicitly enter a header in the first page, there is none (the headers for left and right pages won’t be copied on the first page because independence is requested).
The test Binary (typeset).odt is different. You still have a single Default Page Style but the document is plagued with direct formatting. Among others you manually inserted “ordinary” page breaks and I am not sure about which page variant is activated after these page breaks.
IMHO, you should think more deeply about the logical structure of your document to create the required “independent” page styles for your “independent” parts. Also, avoid direct formatting. You seem to embark for a large document with “internal consistency”. This calls for styles. And Default Paragraph Style is not a style to use routinely: it is there to define your preferred default attributes which are shared among all other styles.