Hi @David ,
Oh, I'm really sorry. I ignored your last sentence and misunderstood your problem. I apologize again.
Your problem is that you want to use anonymous co-authoring in SharePoint 2016 documents. However, in sharepoint, there is no such setting. I searched a lot of documents did a lot of researches. And I helped you consult with our technical support engineers who focus on Office. His reply was that when you start collaborating together, you must log in to your account, so anonymous access cannot be achieved based on co-authoring. And I feel regretful to inform you that it turns out to be a by-design one.
If the answer is helpful, please click "Accept Answer" and kindly upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.