Keep in mind that players have no "Name" field. They have a Nickname, a First name, and a Last name. The "Name" field you see in various places is a field that is constructed, based on the Name Format preference, from the 3 name fields each player has (Nickname, First, and Last).
The pre-defined name formats all will fallback on a non-empty field for a player should the chosen field have no value. So, if you select "Nickname" as the Name Format, players who have an empty Nickname field will have their First name displayed instead. Since the software requires either a Nickname or a First name to be defined for every player, this is guaranteed to not be a blank. Likewise, all of the other predefined formats do something similar.
The custom format will not fallback on anything. So if you specify a Custom name format, and use the string "{NN}", if a player has no Nickname defined then their "name" will be displayed as a blank.
Going back to your original post, you said "A few of my players don't want their real name being displayed on the screens". This begs the question, if a player doesn't want his real name displayed, and he/she doesn't have a Nickname defined, what should be displayed?