I'm just trying to clear up some obvious misconceptions you have about how maps work.
You can't say the flights wouldn't work on a flat Earth, because the Earth is flat and they work just fine. Anyway, there's no map to disprove it, so once again, you can't say it wouldn't work.
This is a bit long, but here goes.
You claim that, "the Earth is flat and they work just fine", but the Earth being flat is far from proven.
Then your statement, "there's no map to disprove it, so once again, you can't say it wouldn't work" is certainly not necessarily true.
It is quite possible to find a set of measurements that will not fit on any flat plane.
On the Globe the following is quite possible, though would need good navigation equipment.
Start at the 0°,-70° (near Japurá - State of Amazonas, Brazil) travel 10,000 km due north to the North Pole,
then turn 90° right (to starboard if you like) and travel due South along longitude 20°E for a distance of 10,000 km
to lat/long 0°,20° (in the jungle on the Equator, Democratic Republic of the Congo),
then turn 90° right again and travel due West along latitude 0° (the Equator) for a distance of 10,000 km.
On the Globe this will get back to the start in Brazil, but it cannot do that on any flat surface.
Another somewhat similar set of measurements is the equatorial circumference of the Globe is very close to 40,000 km
and all Polar circumferences are also very close to 40,000 km. These cannot fit on a plane surface.
Now these measurements are not very easy for individuals to perform and that is where arriving at approximate distances via air route times can give some sort of answer.
These Southern Hemisphere routes make the usual
North Polar AEP map quite untenable.
So
Danang gas suggested a
South Polar AEP map. I can see the attraction for him because it makes the Southern Hemisphere shapes and distances more realistic. He and I are both from south of the Equator.
But, of course, the Beijing to Los Angeles route would make that map quite untenable.
For any flat earth continental layout that you chose, I claim that I can find sets of flights that would be impossible on your flat earth.
What I have not yet done is to get accurate data for a set of flights that
could not fit on any flat surface.
My excuse is simply that there is a lot of work involved in doing it properly - and I am too lazy.
Our newly rechristened
Brotherhood of the Dome has actually done quite a good job for many Northern Hemisphere routes.
He has done a few overland southern ones, and his distance agrees with the actual Perth-Sydney distance within a few percent.
But he refuses to look at intercontinental southern routes because "they are not fully tracked".
Well, that's my story and I'm sticking to it!