People absolutely used to buy (paper) newspapers for the crosswords: they were incredibly sticky and not expensive to make. I don't know whether that still works in an era with tons of free online puzzles, but if the NYT crossword is particularly competent/unusual/good for some reason then I'd not be surprised at all if the crossword sub drove news reading rather than the other way around.

(My experience is more with the British-style cryptic crossword, where every pseudonymous setter has a different 'flavour' and fans have passionate favourite days of the week in a given publication, let alone preferring the Times to the Guardian or whatever. I don't think the American style generates quite such obsessive fandom, but people do love their games...)