What you say is true. Any parent with a mixed race child forced to check a box in their school documents will never check white. There is no privilege there.
I have a couple friends who are the product of interracial marriages. This came up a few years ago. They have always refused to complete the sections of a form that asks about race. If they *have* to complete, they select two.
Think about it. How many half-breeds do you know of who “choose to be white”? If THAT was the “privilege”, Colin, Obama, Ben Jealous (NAACP head!) and others would be “white”, and Rachel Dolezal would be too!