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Map Projection - Why don't we draw things out

I have been reading about different map projections and kind of understand where the problem lies. And also, most of the problem was that the old cartographers did not have any other tool other than their pencil and paper.

Is it possible to just draw out every kilometer, every curve on an actual scale on a very big piece of paper with a scale of say, 1 cm ~ 1 Km? I know we will need a very big paper, but then if we use a computer instead of a paper, then scaling down and up would be simpler right?

I don't think you get what the problem is at all.

The problem is that the earth is curved; It isn't an sphere; It's not an oblate spheroid, it an even more complex shape, called the Geoid.

And whenever you try to go from this Curved shape to a flat plane (like the paper you are talking about) you will have some or the other distortions.

That is why surveyors usually will go with a two dimensional coordinate system, and then plot or map everything in that coordinate system.