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White America is not binary racist/non-racist. There is a broad spectrum ranging from white supremacists on the far right to the truly colorblind. But in the middle of this spectrum are large numbers of people who WANT to be non-racist but can't help viewing things through long-ingrained stereotypes. Those people are generally tired of America's PC culture and worry about non-whites who seem to embrace it. They often feel as if they are constantly being tested to determine if they can meet some standard of anti-racist purity.
Despite Obama's long-forgotten joking about Clinton's dancing ability, most of these people see Obama as part of that hyper-PC culture. Choosing Hillary would not have helped, because she's absolutely part of it.
But Biden? He's a guy just like them. He talks a lot, so he occasionally says stupid things. Racist things. Does that mean he is a racist? It depends on who's administering the purity test. Some would say he's just observant and honest. (Let's see those statistics about doughnut shop ownership...)
The great thing is, Obama already has PC America's support locked up. But he needs to appeal to all those whites whose attitudes on race don't fit into simple racist/non-racist categories. By choosing Biden, he shows them that he understands that world-view. (Consider Tiger Woods. One reason he is so popular is that he's shown these whites that he, too, can joke about fried chicken and watermelon.) Finally, there's absolutely no way that the Obama/Biden ticket can be attacked for Biden's screw-ups by the Republicans; that would contradict the Obama-is-an-elitist (read PC fanatic) plot line than they have based their campaign on.
Of course, that all depends on Democratic voters keeping their eye on the ball, which recent history has shown they do rather poorly. So, to all you angry, betrayed absolutists way out there on the left edge of the spectrum: Stop. Breathe deeply. Remember that it's not about your narrow beliefs. It's about building a strong enough coalition to win. White Americans who have fears and failures about race are not the enemy. Learn to distinguish between the virulent racist and the guy who just doesn't care that much. Stop trying to fix them, and start trying to see what you can do together.
(Of course, all that's easy to say living in the good ol' ROK.)