It would have to be undeniable and a source of financial concern to his conservative Wall Street donors, to the point that they were talking about it openly at the cocktail parties the senior editors/courtiers are allowed to attend.
Otherwise, Joe Kahn and his ilk are going to keep pretending that this is a person who’s fit to hold the office of President. The more he declines, the more pieces we’ll see about MAGAts in Real American™ diners expressing their support for him.
The NYT serves a very specific audience: high net worth individuals who live or once lived in Manhattan or gentrified Brooklyn, and to a lesser extent those realistically working toward that status. Their ideology doesn’t matter to its editors so much as their financial and social class. When you read and appreciate the good, solid reporting that the paper does – it does a lot of that as well as part of its service – and you’re not in that club, keep in mind that that they’re table scraps.