You were correct, webdav did work. For some reason it only lets me open files and not add files, but I am able to see the files on the Alfresco site adding '/alfresco/webdav' to the site URL when mapping the drive, which in my case was https://[mytrial].alfresco.com/alfresco/webdav.
If you open webdav via browser (IE, Chrome or Firefox), you will ONLY be able to browse and download. If you map a webdav network drive in your Windows environment (as another network drives in Windows), you can add files if you have permissions to do it. In Windows, you may have problems to set it up if your Alfresco is not configured for using https.