@sabkan Most likely in your document PageSetup.DifferentFirstPageHeaderFooter. Please reset this option, in this case primary header will be shown on all pages. Please modify your code like this:
com.aspose.words.Document doc = new Document("C:\\Temp\\in.docx");
com.aspose.words.DocumentBuilder builder = new com.aspose.words.DocumentBuilder(doc);
builder.getPageSetup().setDifferentFirstPageHeaderFooter(false);
builder.moveToHeaderFooter(com.aspose.words.HeaderFooterType.FOOTER_PRIMARY);
builder.getFont().setName("Arial");
builder.getFont().setBold(false);
builder.getFont().setSize(10);
builder.write("Footer text ");
builder.insertField(" DATE", "");
doc.save("C:\\Temp\\out.docx");
The document had that setting of first page different on.
On setting like this as you suggested ,setDifferentFirstPageHeaderFooter(false); the issue was fixed