The prevailing opinion on the subreddit I got this from is that the problem is with the word 'privilege' which implies an advantage when the reality is more of a lack of disadvantage. I tend to agree with that as most vitriolic rants against white privilege seem focused on personal struggles, which everyone has, rather than understanding that it means white people in America are generally the vanilla, placeholder, default that no one projects much prejudice onto.
This reminds me of a really nice analogy someone told me about why people get upset when someone "gets more rights."
Let's say everyone in the world gets a cookie. You, a middle class white man (for simplicity's sake), get a nice full chocolate chip cookie. Your neighbour, however, is black, and even though you two are otherwise identical in every way, he only gets half a cookie. This is pretty good, you think. Everyone has one cookie. Seems reasonably fair.
Until someone else comes along and says, hey, that it isn't equal at all. So, they give your black neighbour another half of a cookie. To you, your neighbour has two cookies, and you still only have one. That isn't fair at all! But what our protagonist has failed to realize here is that it's not about how many cookies you have, it's how much cookie you have in total.
I love that intersectionality is gaining traction. The size of your cookie is not your fault, but it is important we can recognize and acknowledge our privilege and help shoulder some responsibility for those who were given a smaller cookie.