The download button shows for any file type as long as the user has read access on the node. So there shouldn't be any customization necessary for that.
How do you want users to get to the documents? If quickshare works for you, what about writing a script that makes the document shared? You could trigger it from a rule, for example. Or, if you want to trigger it from a web script, see this thread.
If you want users to use regular Share to get to the documents, add the EVERYONE group as a Consumer to your site.
If neither of those options sound good, you could write a web script that runs as the administrator and streams the content. This option would completely by-pass security.
Similarly, you could write a small application which would use CMIS or the REST API to fetch the documents using a 'service account'. You would need to grant the service account read access to the folder(s) or content items you want to make available to the application. Your application can then either use its own authentication or no authentication at all. Your application is kind of like a proxy in this respect.
So there are a lot of ways to let users download content without authenticating. It just depends on exactly what you are trying to do.
Hope these ideas helped,