I have tried to use the DAM extension (digital asset management), but that is such a convoluted, under-documented beast, that I got rid of it very quickly. Now I still do not have a solution, but I welcome every hint on how to solve this.
Typo3: Filelinks are not actually links
Whoever named the Typo3 filelinks, should be fired. Out of a cannon. Into the sun. This is because the filelink, does not actually link to a file, but it copies it to some folder (upload/media by default). Why this is so, is beyond my understanding. When you add a simple link in the RTE, and link to a file, you get the expected result: You get an <a> tag linking to the corresponding file in the fileadmin/ folder of Typo3. But when using the filelink, the file will be copied to the upload/media/ directory.