Actually, Peterson does not say “white privilege does not exist”. The point he is making is that there are many different kinds of advantages. Being in the majority is one of them and the majority is white. He essentially agrees with you that it is a bad term.
Now we can debate about where this majority privilege ranks in the scheme of things for African-Americans, who are the minority. Is it more or less important than economic privilege, which is more driven by racism that being a minority? Is it more important than educational privilege, which is more driven by racism than being in the minority? What about attractiveness? And so on.
Peterson doesn’t say there is no advantage to race, though he might rank order race privilege differently.
People love to give examples of white privilege, but more often than not they are giving examples of majority privilege. Sometime they are examples of racism. To use the term white privilege lumps these two together in a way that masks an important difference which is not helping anybody.
Whenever I encounter someone who says there is no such thing as white privilege, I ask them if they think there is any disadvantage to being black. Every once in a while, some white person will say “no” and I simply ask if they woke up tomorrow and they would black, would they be happy about it.
OK, so now we have established that there are at least some forms of disadvantage to being black. All those disadvantages you don’t have by being white are white privilege. Sometimes the response I get is something like, “well, what about affirmative action”? To this I say, “that is a black privilege and it exists because our society generally views that blacks have more disadvantages than whites. It balances the scales a bit”.
So, now we have established there is white privilege and there is black privilege, I make a list of all the black disadvantages. Income, education, crime victims, etc. using real data. And, I ask them to make a list of white disadvantages so we can compare lists.
Maybe I am a shark too, because I have never lost this argument.
I will grant you, however, that starting with the term white privilege doesn’t make it easier.