I think a significant amount of that hostility isn't necessarily denying the existence of privilege, but denying that it's a useful way of framing problems.
I also suspect a lot of it is backlash from over-enthusiastic social justice advocates trying to shoehorn absolutely every social problem imaginable into a context of unilateral power dynamics.
It once was the case that privilege was seen as unilateral and one dimensional. I'm not sure this is the case anymore on the cutting edge of so-called privilege theory.
A black man in the United States might suffer from some effects of white-privilege (vs. white men) while benefiting from some aspects of speaking-English-privilege (vs. recent immigrants).
More generally, I'm not aware of any other framing analysis that is (1) acceptable to anti-feminists and (2) sufficiently nuanced to be useful. Hansonian status analysis is not really capable of providing insight into what we should do to solve the perceived problem - even if it is descriptively accurate (at a high level).
As Multiheaded added, "Personal is Political" stuff like gender relations, etc also may belong here.