The map is a projection. The distances are meant to be accurate along longitude. There is no flat earth map because there are no flat earth cartographers.
There is no need for a cartographer. It would be sufficient for the purposes of these discussions if someone merely sketched out a rough drawing that showed how it is possible to show the size, shape and distances between the continents and major airports in a fashion that accounts for the actual recorded data from tens of thousands of flights as accumulated here:
http://flightaware.com/ including all nonstop flights in the southern hemisphere.
In order to make it even easier, you could cut out the continents from an exiting map that is known for showing them in the correct relative size and shape such as the authragraphic projection:
https://www.wired.com/2016/11/weird-globe-folding-map-isnt-perfect-close/ (Interesting to note that this flat map can be folded into a rough sphere and then even the distances between continents are correct)
So once you have the correct shapes and sizes, the distances between any two points on the same continent would all be roughly correct. Then all you have to do is arrange them on a flat surface of whatever shape would accommodate the idea of a flat earth (who says it has to be a perfectly round disc?), as long as all distances between major airports on different continents was also close to being correct. (Note that this is where the flat (unfolded) version of the authragraph map falls short, not to mention that you would fly over the edge of the world (or at least off of the territory shown on the map) if you head west from Africa or Europe or east from the Americas.)
If the earth were truly flat, this would not be that much harder than a child putting together a jig saw puzzle. The simple fact that after all these years of people debating the shape of the earth, no one has ever managed this simple feat is quite revealing.
Meanwhile the distances and flight paths calculated on a great circle mapper based on the geometry of a sphere all work out to be the same distance and path that the airlines fly on nonstop flights (with a few exceptions based on geopolitics as airlines sometimes tend to avoid flying over war zones). See:
http://www.gcmap.com/So you have one model of a flat earth that no one can even represent on a map equivalent to a child's drawing that still roughly works to account for all distances that are well documented and flown thousands of times a year (there are over 3.5 million commercial airline flights each year, not including military and private flights).
And then you have another model of the earth (the globe and resulting calculators based on a sphere) that when referred to matches up with the extensive database of actual flights and the reports of individuals who fly those routes (see
http://creation.com/a-direct-test-of-the-flat-earth-model-flight-times ).
I wonder what can be concluded from the relative success of these two models, one that has not ever been loosely mapped out even by those who fervently believe in it, and another one that can be used effectively and reliably to show the time it should and does take to fly between any two airports that have nonstop service? Any one want to venture a guess as to why there is such a huge discrepancy between the two models and their ability to be accurately represented?
This was also debated (if you can call it that as I am still waiting for anyone to offer a reasonable rebuttal to most of the points I made) on another thread starting with this post:
https://forum.tfes.org/index.php?topic=5888.msg113568#msg113568