It seems to me that intersectionality makes sense when talking about law. In Crenshaw's example, if I recall correctly, she talks of a company in which black women miss out on their rights because the law only recognised race and sex as protected characteristics. So black women who were discriminated against as black women had to sue for racial discrimination or for sex discrimination. The company could then defend themselves against race discrimination by pointing at all the black men they employed, and against sex discrimination by pointing at all the white women they employed. The way the law was structured made it actually impossible for black women to take action. The law needs to take account of the intersection of two protected characteristics.
But this is a process of creating a tool, recognising intersectionality between protected characteristics, to deal with a clear and identifiable lack in another tool (the law) which has been consciously and intentionally created for a specific purpose. Law is required to be coherent and, well, systematic so that it can be used to decide issues in the business of a variety of different interests, institutions, individuals and groups, as well as people's relationship to the state and state institutions.
But I don't think "society as a whole" has been consciously and intentionally created in the same way. It isn't systematic, though it does contain different systems within it. It comprises a whole bunch of different people who will not agree with each other, will not have exactly the same values and beliefs, and also should not really have to. So when applied to "society as a whole", the theory of intersectionality just creates a totalitarian pecking order that will not allow people to be different and to disagree. It creates a view of the world where there is only ever one right answer, only one viewpoint is correct. That's why it turns into an "Olympics of oppression", as one PP upthread called it.
It's a great ideology for authoritarians, as it justifies coercion as much as any religion ever has and enables control through "divide and rule", such as suggesting that "white feminists" aren't real feminists, pitting women against women. Intersectionality is one of the most anti feminist ideologies out there in the world today because it denies that women can have issues and interests purely as women. This implies that there is no right for women to organise politically as women. They always have to step aside for other people (which, clearly, means men).
The points froggy brought up in her excellent post upthread need to be taken very seriously and dealt with. But they can't be if other people are systematically denigrated for "privilege" (yes, I know froggy isn't doing that. I'm talking generally). The absence of oppression is not a privilege. It's a situation which needs balancing, but there is nothing unjustifiable in not being oppressed.