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.
href, https://enterprise.arcgis.com/en/portal/latest/administer/windows/roles.htm
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.
