It kinda is broad and not meaningful, but not for the reasons you think. "White" now is mostly used to describe light skinned "european-blooded" people, but not even a hundred years ago, Irish and Italian people weren't considered white. And 300 years ago, the only people considered white were anglo saxons (meaning Hitler wouldn't make it into a white ethnostate back then, cause germans were considered swarthy). Race is just a social construct.

At the same time though, the idea of race is something that is culturally ingrained into people from a young age, and is a factor by which discrimination has occured in the past. It was only maybe 50 or 60 years ago when jim crow was abolished. And people still struggle to pull themselves out of the slums because of that. Not even to mention slavery. Generational wealth: If you're poor, your kids are most likely to be poor too. As are their kids. A rarely ending cycle of poverty, and all the baggage that comes with it. And to make matters worse, over 70% of extremist crimes last year were by white supremacists: a position that isn't possible to hold without turning to conspiracy theories about jews and pseudoscience over rationality and reality.

As much as I think society should abandon that schema, it's not happening any time soon, and it's a bad idea to completely ignore it as the delusion of race has a tendancy to hurt people if left unchecked. And we still have people licking their wounds over past offenses who need to be tended to. The privilege in question, is simply that you don't have to know what that entails if you're not a victim or target yourself.