I believe we already answered your question via email today, but for anyone else with a similar problem here's the answer:
The Send Free Download Link function uses the product email message, which can be edited or disabled within each product's settings. On default settings that section is checked and no custom text is entered, which will produce a short default email with a download link whenever that product is ordered or a free link is created.
If that message is turned off a free link can still be generated, but no email will be sent. Alternately, if the message is customized and the default link text disabled an email will be sent but no link will be included.
In this specific case the email was customized and the "Enable Templated Email" box was checked, which disabled our default link text while allowing for the use of special email templates like [%download_link%] -- this option allows the user to fully control the text and where the download link displays, just be sure to include that template in the message so that the link is sent. Alternately unchecking the templated email box will restore our default text and link and include any custom text underneath it.