Jon did a good job at teasing out what I was getting at and restating it in more neutral terms, and I agree with everything that he wrote.
I expect that when people propagated this "white privilege" rhetoric, they had good intentions. They probably thought would be good to get rich white college kids to reflect on the "unearned" advantages that they receive; that it would make them more self-conscious, more empathetic to people unlike them. Perhaps to some extent it does do that. But the flip-side to it is that it also makes white people perceive their own whiteness as a salient identity (or even as the most salient one). And that's a real problem. When people start going around flippantly talking about how "white civilization is boring," inevitably you're going to get some people saying, "well, actually, white people have produced some pretty cool things." You really don't want people saying, "Well, actually, the American founding fathers were white, and that was good." You don't want people claiming these things as parochial "white achievements" -- ideally, you'd diminish the salience of whiteness -- and also, race -- altogether.
And if you tell people that whiteness is their salient identity, and that the Republican party is the party of white people, what then? If you keep challenging them, and berating them, and telling them that they're increasingly becoming irrelevant? I think that's why many people are annoyed about Sarah Jeong being hired. Not because they necessarily find her no noxious, but because they are frustrated because they see the NYT as endorsing her point of view. People find it frustrating, because one can no longer argue that a major institution doesn't find these sorts of views acceptable. Apparently, they're the sorts of opinions that the NYT thinks people need to hear more of (or something)? Her hiring demonstrates something about what the journalistic class takes to be acceptable, uncontroversial speech.
I agree that racism is bad no matter who does it. All of these people who define racism as not just prejudice but "power plus prejudice" get something really wrong, which is that you can talk about "punching up" and "punching down", but those things merely entrench racial hierarchy more deeply.