• Using permission sets you can restrict Mass File Download access for users, Assign a Non-Admin permission set to that user to which you don't want to give all access. 
  • So the user can’t get access to change Mass File Download settings on the About Page and Mass File Download Tab access. 
  • They can only access following Lightning components on the record page where they can download bulk Files/Attachments from the selected record and related records with just one click.
    • MassFileDownloadLightningComponent
    • MassFileDownloadRelatedFilesLightningComponent