Firstly, the plane would only do that if the pilot a) believed in FET, and b) was a allowed to use it for navigation, both of which are incredible unlikely. Secondly, even if the pilot took the most efficient FET route, there should still be a massive disparity between the northern and southern flight times, which simply isn't there.
Substantiate your claims.
By all means:
The vast majority of the world thinks the earth is round, and so it is unlikely that the pilot would believe in FET, and no airlines use FET for navigation. Therefore, the route the planes take would just be a direct one in the RET map, with no circling north.
Secondly, in the FET map the southernmost countries are very far away from each other, being on the rim, while the northernmost countries are close to each other, being near the hub. In the RET map, the southernmost countries are fairly close to each other, being near the south Pole, while the northernmost countries are fairly near each other, being near the north pole. Therefore, if FET was correct, it should take longer to fly between southern countries than it would to fly between northern countries, while of RET was correct, it should take the same time to fly between northern countries as it does southern countries. As shown in the links posted earlier, the flight times between northern countries and between southern countries is nearly the same, showing that RET is correct.