intersectionality is very much in vogue, however the point that isn't made too often is:

Within Race: we discriminate based on gender, class and sexuality
Within Gender, we discriminate based on race, class and sexuality

you get the point

That isn't quite what intersectionality is about… ok, it isn't at all.

Intersectionality is about the fact that people aren't simple beings. For example, often within Queerty, people can be gay or of color, but never the two together, seemingly, and add in gender and class and there you go. Intersectionality seeks to explode us from single-position politics so we can understand how various systems of oppressions come together to create complex, modern monsters — if one will allow.

Intersectionality seeks to include those parts of unstable identities that are left out due to convenience or discomfort… for example, during the SC Democratic Primary debate, Hillary could be a woman, but never white. Obama could be B/black and a man, but never heterosexist (!). And Edwards could be a white man, but the significance was lost because being white doesn't have significance yet on a mainstream level and no one expects him to "apologize" for being a man.

So, your idea of what intersectionality is and your "speaking points" are kinda… well… erroneous. Since intersectionality seeks to speak about individuals caught in the history of institutionalized and social oppresions.