Except that gravity wouldn't always point in the same direction if the Earth was flat. Therefore you are wrong.
If you are at any point upon (or above) this flat earth, then gravity is usually pointing downwards.
Gravity doesn't "follow" the "down" direction.
Gravity DEFINES where is "down".
Down is where things fall.
Down is where we put our legs to counter our weight.
And on Earth it is towards the center of the planet.
What you suggest is what people already try to do when draw maps.
In that process they always have problems, because when you unfold spherical surface to scale it down to flat paper you lose linearity.
Try to cut open ordinary children's plastic ball from Walmart and spread it flat on nearest table.