This narrative has a lot of legs, but also not much of substance to support it. That doesn't mean that feminism hasn't all too often been extremely white centric, but both Susan B. Anthony and Eliz. Cady Stanton were lifelong abolitionists. Stanton spent her honeymoon at an abolition convention in London. They absolutely supported the right of Black women to vote and only opposed the 15th Amendment because they wanted to see all women included as well - not because of racism.
Most early suffrage conventions and such did not invite Black women because it was the late 1800s and there were literally no situations where whites and Blacks mingled socially. But that doesn't mean that they did not support voting rights for all.
"She (Anthony) would not campaign for a federal amendment that enlarged the male electorate and left all women outside the body politic. Speaking to an audience of African-American men in New York City in June 1868, she opined, if voting “be an inalienable right, it is as much the right of the black woman as it is of the white. And you can’t ask it for any class of men, without asking it for all the women who are deprived of it.”
https://www.nps.gov/articles/000/fraught-friendship-susan-b-anthony-and-frederick-douglass.htm
"In 1866, they organized the Eleventh National Women’s Rights Convention, the first since the Civil War began. Unanimously adopting a resolution introduced by Anthony, the convention voted to transform itself into the American Equal Rights Association (AERA), whose purpose was to campaign for the equal rights of all citizens, especially the right of suffrage. The leadership of the new organization included such prominent activists as Lucretia Mott, Lucy Stone, and Frederick Douglass."
https://en.wikipedia.org/wiki/Susan_B._Anthony
There are plenty of things that we can call white women to task for, and I agree with you that we should be doing that, but there is overwhelming evidence that the leaders of the suffrage movement were committed to racial equality in a way that was unprecedented for the time. I'm not sure what the motivations are for this this prevailing story that they weren't, but the evidence doesn't support it.
"Anthony expressed a vision of a racially integrated society that was radical for a time when abolitionists were debating the question of what was to become of the slaves after they were freed, and when people like Abraham Lincoln were calling for African Americans to be shipped to newly established colonies in Africa. In a speech in 1861, Anthony said, "Let us open to the colored man all our schools ... Let us admit him into all our mechanic shops, stores, offices, and lucrative business avocations ... let him rent such pew in the church, and occupy such seat in the theatre ... Extend to him all the rights of Citizenship."
