I don't think anyone comes out of this looking good to be honest. I don't think either side put forward arguments well or in a way that was likely to function as good communication with the other side.
In case Sam comes back, here's an answer that might be a bit more readable than the aggression she encountered. I've actually liked the intersectionalist theories I've read and I suspect some people on here have as well. A big problem with the twittersectionalists people have here is they somehow manage to downplay class constantly (and this point was put to you a bit but quite aggressively). I don't think intersectionalist theory has to be used in that way. And you, Sam, if you ever look back in here, seem to have joined a blogosphere/twittersphere group that probably doesn't want to think about class for a reason: that they are the beneficiaries in that game. You yourself may not be and I'm happy to take your word for that. But you've become part of a little internet scene and there are reasons people here have a problem with that scene. I couldn't necessarily say those people are on the wrong side of the capital-labour divide, but as high-end waged workers they may be in an incredibly comfortable position - and I have been told off recently for 'wrong' views on intersectionalism by a black woman who's parents have a small property empire (true story). I didn't think much of her position I can tell you. I think what she was doing was divisive but did it really *matter* to her as much as to someone who really needs political struggle to have a decent standard of living? I'm sure you are different from some of the other participants in that scene, but you could think a bit more about what you've become part of - why it is that class is not discussed much.
We're (nearly) all being fucked, Sam. Some to differing degrees and in differing ways, but we *are* all being fucked and that is why class arguments matter. That doesn't mean race, gender etc issues should be ignored or that class trumps all. But it's pretty fucking important and this is not reflected much in the twittersphere you are part of.
And the second major point. I don't mind certain statements that could come out of privilege theory. There are people here who are hardline against it but I can see some reason for it. But again where it has led lately in the groups you have been frequenting is deeply problematic. I would have no issue with the statements "White people should listen more to the experiences of black people" or "men should listen more to women." They are general statements that confront some power dynamics in our society. However to move from that to say that, say, a white person should always bow to the opinion of a black person on race issues, or that someone with a certain skin colour does not have a right to an opinion on certain issues, is a brutal and nasty leap of logic. It also does not chime with people's experience, since I'm sure white people here have met black people who say 'I don't think racism is an issue - black people should stop moaning about it' or women who are anti-feminist. But there is more to it than that. The leap from a general statement about power dynamics (as above) to saying that each individual takes on their oppressed-race or oppressed-gender role when they speak on those issues is actually quite horrendous. It is not much better than race essentialism or gender essentialism. It has progressed from those to a 'race experience' essentialism or a 'gender experience' essentialism that ignores the wildly differing experiences people might have, the wildly differing reactions they might have, and ignores all the ways in which people who are oppressed might become complicit in their oppression (perhaps for reasonable, perhaps for selfish reasons) and all the ways in which one can learn to empathise across oppressed groups. It also assumes that you can *spot* the forms of oppression a person is subject to, or even that you can ask them what their oppressions are and get a full answer (you can't).
In addition, we have to develop discourse among ourselves collectively and we are all participants in that. Most of us have had quite enough of being told to shut up by those in power, or being ignored because we don't matter. Having our allies telling us to shut up because of our skin colour is not going to create a struggle we can be part of. Being told to be a bit quieter so others can speak - that I can deal with - being told my opinion on anti-racist struggles doesn't matter if I try to put myself on the lines in that struggle? That doesn't work for me, and nor will it work for millions of other people who you are potentially putting off political action with this line of logic.
Hope that helps. Really.