So, I took a minute to examine the alleged Santiago to Melbourne flight. I say alleged because FEs have tried to book the flight but they get cancelled every time they've tried where they've either had their money refunded or were switched to a 1-stop flight days before the flight.
Let's assume that the Gleason Map is 100% accurate. It's an Azimuthal projection of what they believe to be a globe, so based on that, the Southern Hemisphere is going to be somewhat elongated. But assuming for now that Gleason is 100% accurate, running the geometry, the distance between the two would be roughly 14,000 miles on the Gleason's map. I can show the math but it's pretty simple, since both Melbourne and Santiago are at both between 33 and 35 degrees south latitude, which would put each one around 8,500 miles from the north pole ... given the alleged distance of about 12,000 miles from pole to pole. That would put it at 17,000 miles if one flew from Santiago to the North Pole and then from the North Pole to Melbourne. But the route wouldn't go over the North Pole, but would instead cut across the side a bit. On the Gleasons map, there's about 20 degrees cut off from the 180 degrees hemisphere dividing line, creating a triangle in relation to the North Pole, allowing the calculation on the Gleason's map of about 14,000 miles (vs. the 17,000).
As I said, the Gleason's map is a projection of an assumed globe, is probably closer to reality (for reasons I'll explain later) than other projections, but may be off. To me, South America is tilted a bit too far to the East beneath North America. If it were straightened out a bit, and Australia shifted a bit East, that would significantly cut down the mileage even from he 14,000 calculated there.
Here's Gleason's for visual reference ...

If South America were tilted and straightened a bit more under North America (instead of being "crooked" as depicted in Gleason's), and the size of the Pacific reduced a little bit (as it seems to be stretched out), the distance could easily be closer to 10,000 rather than 14,000 miles.
Standard Airbus planes have an advertised cruising speed of just under 700 MPH. That would allow them to traverse the 10,000 - 11,000 miles in about 14 hours, which is the stated flight time. Meanwhile, the distance between Santiago and Melbourne alleged on a globe is a little over 7,000 miles. That would mean that the Airbus is travelling roughly at 500 MPH (vs. it having a cruising speed of nearly 700). Not sure why they would travel nearly 200 MPH below the plane's advertised cruising speed when they could cut the 14-hour flight down to 10 by taking advantage of the plane's capabilities ... and that would be a huge selling point for the flight.
So, just the difference between the 500MPH and the 700MPH is nearly sufficient to account for the difference between 7,000 miles and, say, 11,000 miles (given a relatively small margin of error on the Gleason's map). Not to mention that it's common for planes to take advantage of jet streams in order to give them up to an extra 100-200 MPH. And, of course, the 700 max cruising speed is just what's "listed" by Airbus, and the plane can probably do more than that. We had the Concorde jet in service for a while that could do double that, close to 1400MPH.
As has been pointed out, nearly all "Southern Hemisphere" to "Southern Hemisphere" flights make an inexplicable stop in the Northern Hemisphere on route to their destination, inexplicable if the earth is a globe.
But let's have another look at the Azimuthal Equidistant map. Here's the thing about those. As you get farther from the central point of the azimuthal projection, everything gets distended and warped, elongated.
So here's an Azimuthal projection map from the South "Pole".

You'll notice how things north of about the Equator are badly distorted to the point that they're almost unrecognizable and are far larger and wider than they appear on something like a standard Mercator map (the one we all know from school).
But now let's take a look at an Azimuthal projection from the North pole.

Hmmm. Strange. Everything apart from Antarctica is very recognizable. You would expect the Southern half of Africa to be as fat/wide as the Northern, the Southern part of South America to be almost as fat/wide as the Northern, but both continents retain their familiar shape.
Ah, but you say, Africa and South America are much bigger than they should be? Apart from the fact that their shapes should be distorted, are they really too big? That perception is due to everyone being acquainted with Mercator. Later, there was a projection called Gall-Peters developed that claimed that the Northern Hemisphere continents were way too big and the Southern way too small ... on the typical Mercator map, and so they developed a projection where they claim that the Southern Hemisphere continents have their true relative size.

In terms of size, this seems very close to the North Pole Azimuthal projection above.
So why is it that the South Pole Azimuthal distorts the Northern Hemisphere badly (to the point of making it unrecognizable) while the North Pole Azimuthal retains the shapes, the outlines, and even the relative sizes of the continents (when taking the Gall-Peters "correction" into account)? North Pole Azimuthal should have the Southern parts of both Africa and South America getting wider and wider until the shapes of those continents would look almost like a square.
What this mean is that the North Pole Azimuthal projection is a close reflection of reality, that the Gleason's map is "closer" to actual reality.
In any case, there are dozens upon dozens of different map "projections" out there, and they vary wildly. So the best we can say is that Gleason's or Azimuthal North Pole projections are "closer" to reality than the flat square maps. But since they're projections off of an assumed globe, they're not going to be 100% accurate. We don't actually know the true distance between Santiago and Melbourne. But, as I pointed out, given the capabilities of those Airbus planes (even without modification) and some inaccuracy on Gleason's, it's entirely doable.
There's all kinds of funny-business going on with cartography. Here's a presentation from Herve Riboni about how even at smaller scales, there's clear distortion on many maps. Herve participated in competitions to sail around the world, and he did sail around the world, and is expert at navigation, maps, etc. He's done a number of presentations on the various distortions being used to hide a Flat Earth and he's become a Flat Earther himself. Here's one presentation about how maps are distorted even on smaller scales, much less for maps of the entire world, which are all distorted due to various "projections".