In AEM 6.5, I as "admin" or another user from the Administrators group started a "move" operation for ~200 assets (only a few of them are published). This async job finishes successfully, but all published assets cannot be republished due to missing permissions so Request to Complete Move Operation Workflow gets triggered. After I approve this action, assets get published and everything is fine, but this workflow should not be triggered because I have all needed permissions to republish assets in the destination after moving them. We even have a custom check if a user is allowed to republish from the destination folder so this move operation could not even be started if I didn't have the needed permissions. In short, the admin started the async move operation and then has to approve republish workflow.
Admin is listed as the initiator of both workflow and moving job so I can be sure that he has all needed permissions to publish an asset.
Some remarks to your observations here:
My advice is:
Hope that helps!