Can the Finder resume a file copy?
No, the Finder does what it does and that is do its best not to leave anything unfinished.  If it can't finish a copy it deletes the partials.  

You could probably hack together a scriptlet that would let you drag and drop files onto it that would then rsync the file somewhere. You could even make it a folder action.  But it would be limited in where the files end up.