They aren’t going to fly the South Pole route, but from Newark they fly left, up, or right depending on the weather.
Even if the polar route was shorter (JFK/EWR-SIN), I think they would be hesitant to use it due to the lack of emergency landing places along the way. (even the Pacific is dotted with landing strips).
The picture I posted only shows the left and right routes, but I’ve seen a report that they flew Canada/North pole in one direction and Pacific-side in the other direction. I think it depends on the wind.
It was the NT, so the real hazard is cattle/water buffalo. When I visited, I was told
Be very careful driving at night: at night, don't drive.
There are flights that use the polar route as normal procedure. For example SFO to Dubai on Emirates Airlines flies almost directly over the North Pole. Nowadays, it’s not that unusual.
Good overview of the seating arrangements when you fly for that long.
A new long flight is in the works. I love some of the seating arrangements, especially the skycouch.
You arrive in NY 15 minutes after leaving NZ. 
The QANTAS Airbus 350s that will be used for these longhaul flights have some interesting features.
I took Emirates Dubai to Toronto, and it followed pretty much the second part of that left hand (Atlantic) track. I have some nice pictures of the southern end of Greenland from the air as well as of Iran and the Caspian Sea.
Wow, those First suites are larger than some hotel rooms I’ve stayed in!
That flight is a popular insiders’ tip among Germans who want to go to New York. You can buy a ticket for the Frankfurt-New York leg and board at Frankfurt. You get Singapore Airlines service (which is outstanding) on a route where you wouldn’t expect it.
Or so I’ve heard; I’ve never done that myself. I have done Singapore-Frankfurt (also my longest flight so far, at a bit more than 12 hours), but I’m not sure if that plane would then keep going to New York.
It’s not non-stop, but it is direct with a refueling stop in NZ. Buenos Aires, to Shanghai - 29 hours.
I wonder why they chose Auckland, of all places, to refuel. Shanghai & Buenos Aires are so close to being antipodal that they could stop for fuel literally anywhere in the world and not add very much distance to the flight.
I suppose it’s because New Zealand is so far out of the way as to offer relatively uncongested air space (and hence cheaper rates than busier airports), while at the same time still being an interesting destination in itself, which makes it attractive for passengers who want to book only one of the two legs of the flight. But that’s just a guess.
Or it could be something as simple as the airline is being subsidized to fly to New Zealand.
Why would it be subsidized? A two hour layover without passengers leaving the plane?
Not every seat on the plane will be taken by passengers wanting the full 29 hour end to end journey.
To make the route viable the carrier is almost certainly going to be counting on China - New Zealand and New Zealand - Argentina passengers. Plus some passengers are likely to want to break the journey.
Whilst the route is a long haul with a fuel stop for the plane, for passengers and crew it still looks like two flights. The plane will almost certainly change crew.
Lots of international flights look like this. Back when I was flying to Europe a lot the standard flight was QF-8 Adelaide - Singapore and swap to QF-10 which was the Brisbane - London flight that refueled in Singapore. It was a tight connection, but a good fraction of the plane’s passengers changed during the stop. The long haul BNE-LHR might have been one of the longer international routes, it didn’t operate as a closed system. I would expect this new route to be the same.
This route is providing New Zealand with routes to Shanghai and Argentina. If there is competition for where the break is, they might provide some incentive to the airline to route their way. New Zealand is a reasonably popular tourist destination, so there are lots of reasons.
There is, or at least there was for a long time, a Singapore Airlines flight from Singapore to New York with a refuelling stop in Frankfurt. For people living in Germany, booking only the second leg of this flight was a popular option to go to New York enjoying high-end service.