When somebody says, "you can't understand because you don't know what it's like to live as a [insert minority] in a [insert majority] world", they're implicitly saying that someone's empathy and/or analytical insight is constrained by their physical relationship to society. That also implies that someone else's capacities in that regard are necessarily broader than your own. And when people make those arguments without even consideration of a particular person's potentially analogous lived experiences--maybe they lived as some other kind of minority (very often a minority in class dimensions)--then there's the implication that a person's character is so completely shaped by their particular lived experiences (as perceived by others) that we can never consider their arguments and contributions independent of that.
There's nothing new about race essentialism. Anglo-American racial concepts are rooted in pseudo-scientific ideas about biological capacities, but that's a very peculiar flavor of race-ist scholarship. There are plenty of white supremacist groups who argue that equitable integration is impossible for various sociological reasons--i.e. racism cannot be completely extirpated and so will always impose an unacceptable burden on all groups in an attempt to achieve the unachievable, so all groups are better off segregated. But they didn't invent these arguments, either.
These same fatalistic sentiments and conclusions (albeit from a much different perspective) can also be found in the writings of famous black authors such as Malcolm X and Frantz Fanon. They're similarly race essentialist--race may be a cultural construct, but it's a durable one that imposes mutually exclusive options for organizing society, similar to class in the Marxist worldview. Frantz Fanon, who grew up under French Colonialism, does an amazing job articulating the various often mundane ways in which living as a racial minority in society leads (necessarily in his view) to systemic oppression of the self and group. These authors likewise believe that a truly multiracial polity is fundamentally impossible. (In Fanon's broader philosophy, racial fault lines lend themselves to class-based oppression in the vein of Marxist class warfare. An observation echoed by modern black scholars like Cornell West and Adolph Reed Jr.--neither of whom is race essentialist, FWIW, though I think alot of West's nuance is lost on people.) In fact, a significant amount of African-American and, more generally, anti-colonialist literature came to these conclusions. And if you speak to many well-read activist black Americans of a certain generation, you'll find many strongly sympathize with these views. Famously Justice Clarence Thomas is of this view--he opposes all affirmative action and many Civil Rights initiatives because he believes that black Americans will never be able to live as unburdened by race as those in white society, and that in many cases these initiatives ultimately increase that inevitable burden. (IOW, Thomas is resigned to a fundamentally unequal America, and that informs his cost/benefit calculus. Thomas' legal opinions are far more peculiar than his racial fatalism, however.)
I found Frantz Fanon quite enlightening, but I don't accept all his premises and his conclusions. I don't accept them intellectually, and as an American I feel obligated to reject them as anathema to our national moral framework--America may be a doomed experiment, but I have a civic duty to believe there's a way forward. My takeaway from Fanon, et al is an acceptance of the very deep, very systemic problems that multiracial polities face (in many ways no less today than 60 years ago); and an acceptance that the answers to those problems are still nowhere to be found. Fanon (better than any other author) not only taught me to recognize the depth and breadth of racism, but convinced me to be extremely wary of claims that it has been lifted. So I accept many of the premises, but not some crucial conclusory ones--e.g. that the absence of answers is evidence of absence. From my perspective it's actually not hard to see how anyone, black, white, or w'ever, could easily make the jump from these intractable dilemmas to the view that we simply cannot ever separate racial identity from our political character. (By political character I mean to say that while we may all be equal in God's eyes, such platitudes have no substantive meaning or application in the real world.)
IMO, we live in an age where we've inherited many concepts from the radical literature of the 1960s and 1970s, but without an appreciation of the context or the consequences. I have no reason to doubt that the radio speaker more-or-less meant what she said regarding her character, and intended it to mean precisely that her perspectives on race (in particular, the burdens and graces of living as a black woman) are intrinsically out of the reach of contemplation by others and therefore of more value in the political debate because of her lived experience as a black person. That is actually a quite common refrain these days. I even sympathize with this. In many ways it's a reaction to white people (or some other majority or power-dominate group--these same dynamics play out in multiethnic societies everywhere) systematically dismissing and discounting claims of racism by rationalizing their inability to see that racism as a consequence of adopting a color-blind mentality. In that vein, I can totally appreciate the use of radical leftist race essentialist concepts from the 1960s to argue that, hey, race and therefore racism is more than skin deep.
OTOH, if you confuse such pragmatic, matter-of-fact arguments with a broader cultural moral philosophy, then you're going to end up precisely where authors like Frantz Fanon did. And unfortunately I think a lot of people do precisely that, without an appreciation for where that ultimately leads. People want to be heard and recognized, and unfortunately sometimes resort to contradictory conceptual frameworks.