First and foremost, IF the earth were flat then it should be easy for FEers to have come up, long ago, with an equally flat map that would be accurate and undistorted in every way, involving no complicated math (only the scale - e.g., one inch equal ten thousand miles), and clearly showing where THE EDGE is. But they can't and won't.
Second, accepting that the Earth is round does not mean that "all flat maps are broken". I think REers will agree that world maps on flat paper would have some distortion problems, and even maps of very large areas such as the whole of the United States or Canada would have some distortions. A map of a city would have distortions also, but so slight that only a micrometer might detect them. Most road maps have worse distortions simply in the width of the lines that represent highways and of the dots that represent towns.
Pilots, navigators, and the like know which maps to use for their needs, how to compensate for the distortions, and so forth.