I don't think you've caught what I was saying. The GP noted indeed why we should expect the distribution of suitably scaled sample means to converge to the Four

I don't think you've caught what I was saying. The GP noted indeed why we should expect the distribution of suitably scaled sample means to converge to the Fourier transform of e^(-x^2/2). I'm saying, it doesn't seem essential to the existence of a CLT type result that this Fourier transform is e^(-x^2/2) itself [i.e., it doesn't seem essential that e^(-x^2/2) happens to be a fixed point of the Fourier transform]; if the Fourier transform of e^(-x^2/2) were some other function, we'd still speak of a Central Limit Theorem and the privileged role of that other function instead. The Fourier transform fixed point phenomenon seems orthogonal to the distribution convergence phenomenon.