First, the NYT is a fiercely anti-Catholic paper and would print anything true or false to discredit Catholicism. Cardinal Burke stands for orthodox Catholicism whereas Pope Francis is claimed by the progressives and political left-wingers so would be more to the paper's liking. Cardinal Burke also is not too concerned with politics.
Bannon is almost totally concerned with politics and after 3 marriages will hardly be too bothered by Catholic Dogma and Doctrine. Of course, he would look for political allies to bolster his clout, however, there is no evidence that Cardinal Burke endorses his political views or is in close contact with him.
Neither is there anything to support your contention that Burke wants to get rid of Pope Francis, there is no mechanism for a start. Many bishops throughout the world have issued guidelines re: AL contrary to what the Pope seems to endorse do they all want to overthrow the Pope? I doubt it,