Notice how the drawn Gaussian doesn't follow the bottom row of binomial coefficients of the Pascal's triangle[1]. The model that assumes that each ball independently falls left or right exactly one unit is limited and doesn't actually describe many real world Galton boards.

[1] https://upload.wikimedia.org/wikipedia/commons/d/d2/GaltonBo...

if the distances between circles approaching another circle is more-or-less the width of the ball, the other balls will not affect it much.

+ The additional perturbations make the situation more random, rather than entirely based on the inconsistencies of the board design.