We should also remember that there are several maps that include two poles, and that distance is not the only determining factor for how long it takes to complete a journey. It could be that the accelerator for the Earth speeds up motion at the rim, or that rotational forces move faster the further out you are. Jet streams are common knowledge.
I don't think distances work properly on the bipolar map either, and I couldn't see the sun and moon paths working on a bipolar map, maybe someone has worked it out.
EA is probably best left to another thread.
I haven't seen anyone who's genuinely tried to analyze which maps are possible, there's typically just a cursory glance at the azimuthal projection that is not claimed to be accurate, with no thought given to subtle rearrangements of the continents. The bipolar possibilities rarely get a look-in, and when they do it's rarely in any detail.
The claim that distances do not work is likely impossible to prove. To try and do so would require proving a number of far trickier propositions.
The one thing I will say is to lose preconceptions. A bipolar map may prove more successful if the cliche of a circular path for the Sun is lost: a figure-8 path would explain midnight suns, and if the Sun's orientation alters it would remain facing the continents.
Regardless, as flight times depend on more than just distance, nothing has been proven.