In some of the arguments from the New Yorker apologists, I get the sense that there is a disconnect in how the representation that was satirized is not just an attack on Michelle Obama, but an attack on black women in general. Much in the way that Hillary Clinton throughout her career has been a feminist symbol because of the way and the why behind how she was attacked, Michelle Obama is attacked in ways related to what she represents. This is why the imagery is so offensive to so many people, regardless of the fact it's satire.
A cartoon of Bush being Cheney's submissive housewife, by the same cartoonist, is offered in defense of the New Yorker cover. But that cartoon is only reflective of Bush himself. Bush isn't seen in some circles as just doing what Cheney wants because of what group, or what race, or what religion he belongs to. This is completely different from the attacks Michelle Obama has sustained.
Hillary Clinton means so much to women throughout this country because when she was put in the national spotlight, she was attacked for being a smart woman. She didn't know her proper role. She was called a 'bitch' by people on radio and TV (in different words sometimes but that was the sentiment), and women who were in the workplace and saw those same attacks on themselves day in and day out related to her. She was called a 'lesbian' for the same reasons that women in the workplace were called that. It wasn't just specific to Hillary Clinton, she was bearing the brunt of a larger narrative, a larger sentiment in American society that resisted the rise of women in the workplace.
Michelle Obama is the same way right now. She is attacked as 'militant' and 'angry' not just because of who she is individually, but because that is a prevalent racist stereotype against black women in general all the time. Black women are stereotyped as scary, and the darker the complexion, the more likely they are to be viewed as scary. The way they wear their hair changes how 'scary' they are. Being emotional is spun differently when it is a man or a woman. And being emotional is spun differently if it is coming from a black woman or a white woman. My mother, who like Michelle Obama worked her way up through adversity (she became a highly successful principal), had to always be careful to fend off being labeled 'angry and militant'. My girlfriend, who is a young black doctor and Ivy league grad like Michelle Obama, has to be careful every day as well.
Much of the caricature of Michelle Obama in the mainstream media is a general bigoted attack against black women. That's why so many people are outraged at the imagery of Michelle Obama on the New Yorker cover. They are offended and find it hurtful, even though it's a satire, because it is a bigoted image that is fought against every day by black women in the workplace throughout America. That isn't the same as Bush drinking from an Iraqi oil drum, or Cheney shaking hands with the devil. Images like that are specific to those individuals. It isn't rooted in something broader about the group they represent.