White privilege absolutely exists, as does 'male privilege', 'health privilege' (mental or physical), 'preference/motivation privilege' (e.g. do you enjoy, or can you at least tolerate, doing things that are financially viable) and a whole host of other inherent advantages that a person can possess from birth, or obtain later in life and hold on to.
Quite literally, 'privilege' (less aggressively known as advantage) is obviously a thing. Even if you don't consider the 'white' variant, people win and lose the game of society based on pretty arbitrary attributes.
We generally don't run around speaking about those issues in those terms, though, because it's just not a very tactful way of addressing them (attacking core components of a person's ego), it's much more politically manageable to discuss equality/empathy/representation for example.
It would be equally unpopular to talk about e.g. the 'transgender plight', despite the struggles trans individuals face. Words are important.