Absolutely. And they have. Shaun King has a huge following and there is a lot of questions about his true race. His mom is white, his dad is mixed or light skinned according to him, but some black people from his hometown area say his dad is white (the dad listed on his birth certificate is white, but he claims that's not his real dad...i don't think his mom has ever publicly addressed it nor should she have to).
There is a lot of trust that has to be rebuilt before the black comminuty will have enough faith in white people to be "leaders" of their movement. For now, the best thing for white people to do is support the black community AND listen...and by listen I mean listen without feeling defensive or needing to say "ya but".