Missing the point. Inquisitive is the one questioning distances at all, not me.
Then why do you keep responding if you don't have any answers?
Are you claiming that the distance between the two would be different on a flat earth than it is on a spherical one?
If spherical geometry is very different from planar geometry, then why would you expect the distances to be the same?
As the distances would be different and we know exact distances between places that are on a spherical earth it is not possible to draw a flat earth map using verified data.
Falklands to Australia is probably the best example, it is not via the North Pole.
If you contend that flights from the falklands to Australia would travel across the north pole on the monopolar map, that's just silly. Navigation uses the lines of latitude and longitude, because that's the way instruments work. To verify this, use a compass. Moreover, very few flights travel over the north pole even in the northern hemidisc. It seems ridiculous to want to travel to two southern hemidisc locations via the north pole. Also, since you refuse to research your own question, I went and did a search for flights from Sydney to MPD in the Falklands, and couldn't find any evidence that any airline actually makes that flight, so your hypothetical route distance has no grounds.