[tfehring]

The normal density function does fall out of proofs of the CLT. Usually those proofs stop at either the characteristic function or the moment-generating function of the normal distribution. From a proof that results in the moment-generating function of the limiting distribution [0], you can derive the normal density via an inverse Laplace transform [1]. You can probably do the same by deriving the characteristic function of the limiting distribution and taking the inverse Fourier transform, but I've never seen that proof.

There many ways to derive the normal distribution from first principles, depending on which first principles you start with, because it has many useful properties and shows up in many situations.

[0] https://www.stat.berkeley.edu/~mlugo/stat134-f11/clt-proof.p...

[1] https://math.stackexchange.com/questions/1528845/moment-gene...