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).
Speaking for myself, I find the privilege framework to be the one lacking in nuance or pragmatic application. I use the term "unilateral" because its core mechanic appears to be "person A has power person B doesn't, and person B suffers as a result". Coming from a game-theoretic perspective, which routinely deals with unexpected and perverse outcomes from agents being given different sets of choices, this seems crude in the extreme.
On the subject of multidimensionality, I've read up on intersectionality in good faith, and made an effo...
As Multiheaded added, "Personal is Political" stuff like gender relations, etc also may belong here.