Hi Stephan G,
According to your description, the existing documents got an ID while the new documents couldnt get an ID, but the new document sets could get.
I have tested for many times. However, my documents and document sets could get an ID normally.
I wonder if the related timer jobs ran after you created your documents.
If you want to get the ID immediately, you could try the steps below
1. Open the Central Administration.
2. Click monitoring, then click review job definitions.
3. You could find two jobs called Document ID assignment job and Document ID enable/disable job, click them.
4. Finally, click run now.
For more details, please refer to the link
https://www.webucator.com/tutorial/advanced-microsoft-sharepoint/document-id-service.cfm
Hope it will be helpful to you.
Best Re