Pages does not auto-update Date & Time from the Insert menu in documents or templates.
You can open a new document from that existing template, and replace that Date & Time entry with a single table cell containing the function now(). Then, remove all of the table styling from the selected table's Table panel. Then save the document as a new template.
When you open the revised template, the current Date & Time (depending on cell formatting) will appear from the table's now function. This will only update to the current time each time the document is opened if performing several edits.
If you export this document to Word, you will see a dialog on export that some values couldn't be exported and will retain their last setting. A Finder Quick Look of that Word document shows the last Date & Time that appeared in the Pages document before the export. Opening that exported Word document in LibreOffice Writer behaves the same way. I don't have Word v16.100 on this Mac to test behavior in that application.