I personally don't see this as the strength of the intersectionality argument. To be clear, I am not saying that your point is incorrect. I agree with you completely.

My point is that to people not familiar with the concepts of intersectionality might find this to be a bit of a flimsy position. I could be wrong and I am willing to discuss why in more detail if you like.

I personally think the incompleteness argument is more persuasive. Proponents of individual movements are already aquatinted with the inequalities of their own individual movements and getting them to recognize what their movement lacks is easier than getting people to recognize that integrating other movements will produce better results.

It also, explicitly, forces people to "say the quiet part out loud". They don't get to hide behind platitudes. I.e. the j.k Rowlings of the feminist movements get to out themselves as terfs or racists that much easier. When you get someone to admit that there are more inequalities faced by black woman than either movement alone covers and their reaction isn't immediate recognition of the problem... You have found your problem.