Good Afternoon. I am trying to hide Server Manager folders in order to prevent users from seeing them when they publish to a Server Manager folder. These folders are in the server manager when I go to https://gisserver.domain.com/server/manager and listed under the (site root).
The answer would be "You can't do that for Publisher and Administrator roles".
According to Modify permissions for a service or folder
When a role has its role type set to Administrator or Publisher, members of that role can access all services hosted on an ArcGIS Server site. This implicit permission cannot be overridden by changing the permissions on a service or folder.
Presuming your ArcGIS Enterprise deployment includes Portal based on existence of Hosted folder in your screen capture of server manager? To achieve your goal, federate your Server to Portal and authenticate your users in Portal. Then you may take advantage of finer grain access control afforded by user type, role and groups. Even though your users may see the folder, they will be presented with sign in dialog if they try to access a folder that contains limited access content. If the content is not shared with that named user, they'll either be denied access to the folder or see an empty folder after sign in.
NOTICE: If you follow this method and define a custom role with one or more administrative permissions, then users assigned to this role WILL have access to all services.