In order for Full Name to be public both the Full Name field and at least one name field (Title 1, Title 2, First Name, Middle Name, Last Name, Suffix 1, Suffix 2) must be Public by Default.

To make the field public by default:
  1. Go to Administration, Field Options
  2. Mark the Public by Default checkbox for Full Name and one of Title 1, Title 2, First Name, Middle Name, Last Name, Suffix 1, or Suffix 2
  3. Click Save