After further reading in the FAQ you linked to, I finally dug up this section:
What happens when my subscription ends? How can I continue to work with my documents?
As the expiration date approaches, you will receive notifications in the Office applications and via email, alerting you to the upcoming expiration. If you choose to let your subscription expire, the Office software applications enter read-only mode, which means that you can view or print documents, but you can’t create new documents or edit existing documents.
To return to full Office functionality, you can purchase a new version of Office by visiting www.office365.com or any participating retailer. You could also return to full functionality using an older version of Office, or you can use Office Online for free for basic editing.
So, the locally installed software should stay installed, but its usefulness will be minimal. If you do purchase a 2016 product key, then it should all return to normal, local, functionality. Obviously the online storage will not be available anymore, but you already knew that part.