The question is posed nicely in a letter quoted in the piece:
“My question is what role the paper’s hard-news coverage should play with regard to false statements – by candidates or by others. In general, the Times sets its documentation of falsehoods in articles apart from its primary coverage. If the newspaper’s overarching goal is truth, oughtn’t the truth be embedded in its principal stories? In other words, if a candidate repeatedly utters an outright falsehood (I leave aside ambiguous implications), shouldn’t the Times’s coverage nail it right at the point where the article quotes it?”
Arthur S. Brisbane, the Public Editor, responds by asking:
Is that the prevailing view? And if so, how can The Times do this in a way that is objective and fair? Is it possible to be objective and fair when the reporter is choosing to correct one fact over another? Are there other problems that The Times would face that I haven’t mentioned here?
Here's a suggestion: Budget some money for fact-checking, whether by dedicated fact-checkers or the reporters themselves. And then, make sure every piece of the story that makes a factual claim -- whether it is in the reporter's background or analysis, or in a direct quotation from someone else -- is checked against the available facts. Tell us whether the claims are supported by the available evidence. Present the readers with the facts as best they can be established right there in the story.
Because people reach for newspapers to get factual details of things happening in the actual world we're trying to share. If the paper of record views getting the facts right as a style choice, where the hell is the public supposed to get the facts?