If you are not renaming the new file to be the same name as the origianl file, you can use the delete file server behavior to delete the original file, just reference the hidden form element that contains the default file value.
True enough, there is no need to delete the original file if you want the same file name, but the behaviour is there (in case you want a new file with a new name) and it is triggered off of a successful upload so it will delete it anyways.
It is not unreasonable to require the form to do both. A newly uploaded document may be a correction or it could be a later version of the same document.
So what is needed is some logic that says IF you are NOT overwriting, then delete original.