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The azimuthal projection plots the surface of a globular Earth onto a flat plane. It is not a flat world model.
Flight maps are not modeled on flat earth maps, they are projected onto an equidistant projection for modeling flight schedules because pilots aren't taking little globes with them.

All maps contain distortion, different maps are used for different reasons. The maritime industry uses different projections such as Cylindrical Projection like the Mercator projection. Just because this is a common model and widely used, that doesn't mean the world is a toilet roll according to sailors.

The azimuthal equidistant projection preserves both distance and direction from the central point. This is important on long-haul flights hence the choice for this projection type on modeling in the aviation industry.

Do you understand the difference in flight path distance between the 1st and 2nd picture?