It seems to me like you're describing exactly the problem intersectionality describes; if you go back to Crenshaw's paper, it's just a series of cases where Black women were excluded as class representatives because the fact of their belonging in two separate disfavored categories meant that they wouldn't be accepted in either: they couldn't represent Black people because they were women and there was a male-female pay disparity, for instance, and they couldn't represent women because white women made more than Black men.
What you're saying seems to reinforce that case, not rebut it.
But if you respond to that by having a white man exercise his privilege contrary to what black people as a group would do and what women as a group would do, then you’re engaged in tokenism and reinforcing white male supremacy.
Put differently, the dominant group asserting its power to address perceived problems within those groups—according to the dominant group’s view of justice and order—necessarily disempowers those groups. And it does so in a particularly gross way, by co-opting those groups’ identities and installing representatives those groups wouldn’t have selected for themselves. You can’t meaningfully empower minority groups as groups if white post-Christians retain overarching authority to enforce their view of how those groups should operate.
As an aside, while I love to rag on Kamala Harris, Bobby Jindal and Nikki Haley have the nearly identical problem—their conversion to Christianity. The difference is that conservatives don’t play identity politics so neither Haley nor Jindal purport to “represent” anyone.
What you're saying seems to reinforce that case, not rebut it.