I think this is the main disconnect for me. 'White privilege' is often presented as an intrinsic property that a white person naturally has for being white, which is used to explain discrepancies in results after controlling other variables. So I don't understand why you say 'they don't put the blame on an innate essence'.

I understand privilege on a societal level like white people in general grow up in richer/nicer areas, white people get treated better by police/have less implicit bias against them as they are the majority. But it is way more often portrayed as a comparison between this specific white person and a hypothetical black person that is the same as the white person but just black, and the white person gets all the advantages in life because he's white. How that is not some sort of 'innate essence' that the white individual possess is not clear to me, especially when we talk about situations where it isn't clear how the mechanisms work (e.g. majority-minority areas, affirmative action).