A few thoughts on Clinton's "intersectional" tweet:
1. The immediate response to this was ridicule, but it's worth noting how much more substantive this is than the "intersectionality" of popular liberalism. In that discourse, the concept generally plays one of two roles. Broadly, it's just an empty superlative that means something like "sophisticated" or "good": to say that one's "analysis of oppression is intersectional, groundbreaking, and cutting-edge" is just to pat her on the back. Often, the word is also used in a more specific sense to mean "thorough and correct in a way that Marxist discourse is not". So when one says that "Clinton's understanding of race is intersectional" one often just means that Clinton is right in some unspecified way that, while ostensibly accounting for a class analysis, just-so-happens to contradict it. In popular liberalism, this is what intersectionality has devolved to: vague praise, and/or vague dismissal of class-analysis. You will almost never see liberals articulate or defend it any further.
2. Clinton's tweet, meanwhile, is at least talking about something. She's just listing out a grab bag of problems and saying that they're all related, which is true enough, I guess. There are some basic conceptual problems here if you try to extend the analysis any further - for instance, every node on this network is a symptom of systematic racism, so it's odd to include "systematic racism" as its own node - but if you don't think too hard about it, the chart isn't meaningless or nonsensical so much as it's utterly banal.
3. That said, none of this corresponds with the intellectual tradition it's laying claim to: intersectionalism as a school of analysis developed and articulated by generations of iconic leftist scholars and activists. The entire point of that analysis is to think about various and diverse issues of power and oppression systematically. You are not making the economic determinist move of reducing all such issues to matters of class, but you are still trying to build a simplified framework for understanding the dizzying complexity of the world. So here, for example, instead of just dropping a laundry list of problems afflicting communities of color, an intersectional analysis could involve a chart that looks something like this:
The point here is that an intersectional analysis looks at the way underlying factors like class and race intersect to create problems. There is no real systematic way to talk about the way that problems in general interact, except, as Clinton's chart (helpfully?) points out, to note that they do.
4. In practice, of course, the point of Clinton's tweet was not to lay out some rigorous or defensible analysis of power and oppression; it was meant to win the approval of Twitter users who have a trivial understanding of intersectionality resembling what's laid out in point one. This is a hilarious indictment of her supporters - not only that they would be impressed by this, but that the Clinton campaign knows they would be impressed by this, and is willing to build its PR efforts around such vapid signaling.