Maps of small areas have minor inaccuracies, close enough for practical use. Maps of the entire world that are flat exist in many forms, mercator projection, etc. None can be continuous with correct direction, distance, and constant scale. They are still useful for things that do not require precision.
What you can't do is make a map of the entire world with correct distance, direction, and constant scale. If you have one, please show us.
If the earth is flat, it should be a simple matter to take the distances, scale them, and draw the map on a flat sheet of paper. There are several attempts to do this in the FAQ. All of them show Australia as wider than US.
As for the cover up, interested to hear your idea, but it is not clear from the OP. Not clear what the connection is from animal icon to suppression of FE. You gave a reason as distrust of govt. That is not a reason to cover, that is a reason why people believe FE.
Suppose I am a govt official who knows the earth is flat. Why would I cover it up?
Why do governments cover up flat earth?