As you have anonymous site you are using on Prem. So you are using either SP Designer 2010 or 2013 Workflow. If I remember correctly, that WF will trigger under the account you published it. So you need to publish it with account that has the necessary privileges and set it up to trigger when item is created/modified.
For anyone looking at this in the future, the checkbox mentioned in the related question linked above worked to prevent emails from being sent to users when directories were shared with them. This was done via Classic View on SharePoint Online.