The Fourier transform of a uniform distribution is the sinc function which looks like a quadratic locally around 0. Convolution to multiplication is how the quadratic goes from downstairs to upstairs, giving the Gaussian.

Widths of different uniform distributions along with different centers all still have a quadratic center, so the above argument only needs to be minimally changed.

The added bonus is that if the (1-w^2)^n is replaced by (1-w^a)^n, you can sort of see how to get at the Levy stable distribution (see the characteristic function definition [0]).

The point is that this gives a simple, high-level motivation as to why it's so common. Aside from seeing this flavor of proof in "An Invitation to Modern Number Theory" [1], I haven't really seen it elsewhere (though, to be fair, I'm not a mathematician). I also have never heard the connection of this method to the Levy stable distributions but for someone communicating it to me personally.

I disagree about the audience for Quanta. They tend to be exposed to higher level concepts even if they don't have a lot of in depth experience with them.

[0] https://en.wikipedia.org/wiki/Stable_distribution#Parametriz...

[1] https://www.amazon.com/Invitation-Modern-Number-Theory/dp/06...