Hi Jeff,
I'm guessing you upgraded from a fairly old version, I do recall that link icon being there in the early version of the newer file manager but it was removed exactly to avoid problems like this because it was never intended that anyone would link to files directly in the file manager from content. The file manager is locked down for security and should not be used for anything but uploading and downloading and managing files. To link to a file you should use the link icon in the editor which allows browsing files that you can link to.
The fileservice for security requires a security token to be passed in the url to the file and the security token changes periodically (whenever the application restarts) and it also can only be used by users in roles allowed to use file manager so it is not designed at all to serve files for content links which may be coming from anonymous users. It was an oversight in the early implementation because that link icon was part of the file manager Ui which we got elsewhere and integrated, once we realized that people would use those links we removed it, but apparently you have been using an older version for a long time that included it.
So it is intended and the workaround is to re-link to the files using the link browser in the editor.
Hope it helps,
Joe