This needs clarification IMO. You're missing the word "some" here. There are many reasons people play the race card. Many times it's because it's actually racism. I can appreciate the speech and I can appreciate that it applies to some people. However, it is NOT "the" reason.
Read Dr. Shelby Steele's "White Guilt". The use of the word "some" isn't necessary. Here is Steele on the race card:
"Well, in many ways I think that’s what happened in America. Whites became stigmatized as racist. And from that point on whites were in the position of forever having to prove the negative, that they’re not a racist. And again, if they don’t prove the negative, then the stigma sticks. Well, you must be a racist. And so since that time whites, and particularly, again, institutions, have lived under threat of stigmatization.
Why’s that important? It’s important because if you are stigmatized as a racist in American society—an institution, let’s say. It’s easier, I think, to see on the institutional level. Then that institution becomes illegitimate. It loses its legitimacy. It loses its ability to really function in this society. So the stigma, again, has a powerful impact, because it has so much control over legitimacy. How can you be a legitimate institution in a multi-racial society that is supposed to be free, and everyone is supposed to be equal under the law—how can you be legitimate if you don’t have any blacks in your institution?
So we could look at a disparity like that—blacks could—again, moral authority having passed to us. We could look at a disparity like that, and we could say you don’t have any blacks in your institution. It’s a racist institution. It’s illegitimate. We will sue it. We will do whatever it takes. And in a sense since the ‘60s that’s pretty much what has been —what has happened is that minorities have begun to sort of manipulate that stigma. We call it some circles today the race card. Play the race card. What does the race card mean? Well, if you don’t do what I want you to do, then you’re going to be stigmatized as a racist, and the price you’ll pay is you’ll lose your legitimacy."
Steele acknowledges that racism exists but he treats it as powerless. Steele on racism:
"Now, we had just defeated racism [talking about the civil rights movement]. We had just stepped into much greater freedom. But we were never as obsessed with racism when we were actually segregated as we are today in freedom. Today we are obsessed—we worship at the church of racism. Racism is everywhere, it’s just so subtle. It’s in every little situation. And we’re still being victimized by it, and that explains why we haven’t done better.
And so we are a people traumatized not by oppression, but by freedom. And the great challenge that faces us a people—and I have every faith that we will live up to it, is to learn to thrive in freedom.
We’re free. It doesn’t matter that there’s still racism in the world. White supremacy no longer has any authority. So what if somebody believes white people are superior? I don’t care. If they can’t interrupt my life, and if they can’t keep me from becoming educated and taking out loans in banks and pursuing my dreams, what do I care? I’m not going to lose any sleep over someone like that."
His argument is that it isn't racism that holds a black person back. It's a combination of "bad faith" (ie, excuse making) and "white guilt" policies like welfare that have destroyed black families.