The Instagram story-esque design is an interesting take. I'm viewing this on a desktop but I'm not sure how I feel about it. Usually I prefer scrolling but I suppose it works well with the high-res photos.
I feel the opposite. Rather than read a story I read a bunch of disconnected snippets, like tweets. The separation was magnified by the need to click for each snippet. It's hard to remember what I read.
Once I figured out that it was a slideshow and not a broken page I liked the format. They should have a highly visible arrow to the right though, since scrolling and pressing the down key do nothing.
NYT like other media companies are expermimenting with different formats. Some are more interactive than others. I personally am not too crazy about this one because at least on my phone it appears as if poorly formatted and when swiping a slide back activates to the browser back command.
When at work I noticed that I almost always avoid these types of format especially when it take over the whole screen and have conspicuous animations.