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I dislike most nicknames as given names. Sadie might be an exception, but I can see problems with that as the child ages. Quite often when people see a name like that they assume it is short for something else(Sarah? Mercedes?). Yes, it is wrong to assume, but it happens.
Aside from that, you give a child more options when you give them a whole name. And yes, I do consider most nns to be somewhat incomplete. (There are exceptions, though none come to mind right now.) By giving a longer full name you allow for that child to grow into he or she is, not just who you think he or she will be, if that makes sense. A full name with a nn lets teh person decide who he or she is or wants to be. You may have a Katherine called Katie, but she may rather be Kat or Kitty or Katherine.
Also the ability to use a nn lets the kiddo determine the closeness of the relationship. Only my absolutely oldest, closest friends can call me Kristy with out being corrected that that is not in fact my name. On that note I would also say that introducing her as Arielle, even when calling her Ella, gives her the power to determine what she is called by whom, and would be perfectly reasonable.