First, to address the most basic example of Obama and the conservative doctor mentioned at the end, yes it's okay for the President to use the President's website to promote the President's agenda. And I'm not sure how Obama's race somehow indicts the entire left on this issue (speaking of "painting with broad strokes").
But more importantly, rarely, if ever, have I ever heard anyone on the left characterize simple disagreement as racism or anything else. It's the form the disagreement takes.
So, conservatives, the question is NOT are your views inherently racist or misogynistic, the question is do you present those views and your disagreement with progressive ideas in an overtly racist or discriminatory way? Too often the answer is yes.