Wow, completely agree about how the layout, and emphasis on different headlines, topics, categories, advertising etc.. are all huge indicators of society's interests and conceptions of the world. That said, I would be surprised to learn that the nyt doesn't store their front page every day into some archive? If they don't, then someone must!
The CMS has all the rankings. Rank meta data is stored with articles, including (if published in print) where it was in the physical paper and what editions(s).
On the web site front, there are internal process capturing the Homepage HTML and that goes back maybe 10 years at this point.
On a more public level you can still find the Homepage from a given date here:
