. . . the implication that there is no "no one objecting" to the countless "rap/hip-hop songs which portray women as sexual objects" is absurd. Google misogyny of rap music and you'll find that there's been a long, thoughtful, multifaceted compelling dialog that includes objection, criticism and analysis of the phenomena.
And as always. . . when confronted with a criticism, pointing and yelling "But nobody is criticizing THOSE things over THERE" doesn't negate the initial criticism. If anything, it's a call to broaden the dialog.
That said, I enjoy "Baby it's Cold Outside" and appreciate the interpretation of it above. But your logic, and inaccurate observations of other genres fall short.