Permission Issues for User 1
I don't think Real Name does what you think it does.  At least not by default.  The module is primarily for using parts of the user account other than the username when the user account is displayed (such as the by-line on an article).

You can set the edit link to send people to mysite.org/user/my-real-name/edit but unless you've done something to make that URL exist, which I'm not sure you can do with this module, it won't do anything.