Do you also think of e.g., Fox News as a Democratic media institution, since they're always wall-to-wall criticizing Democrats while going easy on Republicans?

No, I think the tone of Fox is quite different! I don't think the Times is anti-Harvard; it's so Harvard-identified it felt obligated to take down the president of Harvard in the belief that it would immunize Harvard from the kinds of right-wing attacks the Times was abetting.