Southern Hemisphere Flight TimesFlight HopsThe vast majority of flights between the two extremities of the world don't travel between New Zealand and South America directly, but make a bunch of little hops along the continents, picking up and dropping off passengers along the way. Like international shipping, the post office doesn't charter a direct flight just for one parcel of mail. The post office puts it on a flight that makes dozens upon dozens of little hops along the continents of the world, picking up and dropping off mail by the ton, distributing to major regional centers. It is vastly cheaper for the airline company and post offices to operate in this fashion.
With this considered, judging the flight time between two areas would be close to impossible unless you could find a non-stop flight between two distant areas on the Flat Earth.
Jet StreamsFor the rare flight that does occur between those two extremities of the earth; along the courses of international flights between points in the Southern Hemisphere exist enormous jet streams located at around 35,000 ft above the surface of the Earth. In the Southern Hemisphere there are both eastward and westward jet streams. By necessity, in relation to the polar front and subtropical jets, easterly jet streams in the Southern Hemisphere propagate at rates even faster than the westward ones. The winds in a jet stream regularly reaches speeds of over 400 miles per hour. Airliners use these jet streams to faster reach a destination.
Here's a reference from Cubanology.com:
"The Jet Stream is an amazing phenomenon of wind current which twirls around our planet, capturing and changing anything in its path. These giant, so-called "river" of winds can be several hundred miles wide, 1 to 2 miles in depth and can be found from as low as 12,000 to as high as 80,000 feet above the surface of the Earth and regularly reach the speeds of 400 miles an hour."
This variable contributes significantly to the travel time between Argentina and New Zealand. While an airplane may have a rated cruise speed it can provide under its own power, the extreme winds of the jet stream adds enormously to the actual velocity of the plane over the surface of the Earth.
For every moment the plane stays in the jet stream it is being accelerated until it matches the surrounding velocity and momentum of the air around it. From there the plane can use the limits of its power to travel faster than the jet stream. Thus, six hours of travel time between Argentina and New Zealand becomes extremely possible.
In general, the Flat Earth Society has found that winds are strongest just under the tropopause. If two air masses of different temperatures meet, the resulting pressure difference (which causes winds in FET) is highest along the interface.
These facts are a consequence of the thermal wind relation. The balance of forces on an atmospheric parcel in the vertical direction is primarily between the pressure gradient and the force of gravitation by acceleration is a balance referred to as hydrostatic. In the horizontal, the dominant balance outside of the tropics is between the Coriolis effect and the pressure gradient is a balance referred to as geostrophic.
Given both hydrostatic and geostrophic balance, the Flat Earth Society derives the thermal wind relation: the vertical derivative of the horizontal wind is proportional to the horizontal temperature gradient. The sense of the relation is such that temperatures decreasing polewards implies that winds develop a larger eastward component as one moves upwards. Therefore, the strong eastward moving jet streams are in part a simple consequence of the fact that the equator is warmer than the poles.
Flight DelaysOne additional variable we must consider; if you've ever been to a major airport you would know that it seems as if every other flight is delayed. A flight rarely arrives at its estimated time. Most people with experience in air travel, who've been on international flights, can attest to the extreme delays of flight arrivals. If we search for "flight was delayed" on Google we get millions of hits. Reading the direct personal testaments, excluding mechanical failures and bomb threats, we can see that it's not uncommon for flights to be delayed on the range of hours.
Airports may blame the cause to wind conditions, but it is very possible that the maps themselves are inaccurate, especially in the Southern Hemisphere, since they are based on the supposition of a Round Earth.
Just because LAN.com or a travel agent company may measure distances on a Round Earth map and estimate that a flight might take 6 hours, it doesn't mean that it will. Observational and personal evidence will be necessary to back any flight time claims between extremities of the Flat Earth. If there were any complaints due to travel time, the customer relations person at Delta would hardly be in a position to redraw a map and change the shape of the earth. There would be nothing the Delta agent could do, other than blame any delay on wind or weather conditions.