This is a super soft question in a way, but I was wondering why the central limit theorem says that the distribution of means of converges to the normal distribution. I understand why we might want a symmetric distribution, and I understand why we might want the majority of mass centered at 0 (these are both elements of unbiased means). What I don't get is why the specific pdf we converge to is the normal distribution, and not some other "almost-gaussian" (symmetric, centered at zero) distribution?
This is mainly motivated by the fact that Lagrange's proof of the standard Gaussian integral seems kind of miraculous, but the general strategy seems to be creating a radially symmetric product measure from a symmetric measure in one dimension by taking the measure product with itself. The problem is that, on a calculus level it seems like the normal distribution is kinda the only one Lagrange's proof will apply to, but on the measure theory level I don't really see why this is true. It seems that, a priori, there might be any number of symmetric distributions who's product measure with itself would be radially symmetric. My point is just that... the normal distribution doesn't seem that special, except for the CLT, which makes me think there is something special about it. I just don't get why it has to be the normal and not an almost-normal distribution, except, of course, for the fact that it is the case.
I guess my question is pretty simple, why not just an almost normal?