I've lived in working class neighbourhoods for the majority of my adult life, and there are so many observations that fly in the face of white privilege as a theory.
Most of the poorest people I would encounter were white. The people most likely to be long-term unemployed were white. The people whose children were least likely to get an education and escape poverty were white. Almost all of the junkies were white. Every single homeless person was white, no exceptions.
Whether or not HR staff hiring for a white collar job prefer white names is irrelevant when you and the people you are competing against didn't complete high school.
What the Daily Mail think about Meghan Markle or Islam is less important that what the Vietnamese fruit shop owner thinks of you because you share phenotypal characteristics with the homeless ice addict that sleeps three doors down and scares away customers.
This doesn't even start to cover ethnic tensions (Serb vs. Croat, Indian vs. Pakistani atc.) which exist with no regard for the typical sociological black/white/asian racial boundaries and can cause real violence and discrimination.
I will admit that there are racial groups that have it rough simply because of their skin (black people in America, Aboriginals in Australia, Gypsies in Europe), but the idea that being white grants you privileges over and above all other racial groups ignores the lives and experiences of an enormous segment of the population.
(Just to clarify, I do accept that John Smith has an unfair advantage over Rajesh Kumar when applying for a software job at a FAANG. I also admit that this has worked out well for me. This doesn't extrapolate to the whole population, though; many white people will experience little to no privilege because of their race.)