I don't dispute that; but 16% of a larger grouping of the 86% of the population is still a huge amount of children. So whilst there may be some privilege with being white - it isn't univerally shared - and certainly not shared amongst the traveller communities (who are also white).
That's the point of the article. White privilege, as an argument doesn't feed these children - and the methodologies being used to improve the academic achievement of other ethnic groups isn't working in the same way for white poverty stricken children. Why? Plenty of reasons. There may be advantages that come with being from a religious group, from living in cities with large amounts of local authority money. Coming from a place where your entire family is based. These things aren't specifically to do with skin colour, but they are sometimes to do with demographics.
Your view is they already have enough privilege... but maybe this privilege thing is a bit more complicated and we shouldn't entirely focus on skin colour and take a more holistic view. Your view will drive poor white people to the political right. That's the long term issue we face in this country and the US will have to deal with too.