Yes. A NN consisting of, for example, sigmoids will form an “algebra” that separates points in the sense of the Stone Weierstrass theorem, and the NN approxima

Yes. A NN consisting of, for example, sigmoids will form an “algebra” that separates points in the sense of the Stone Weierstrass theorem, and the NN approximation result will follow. All that’s really needed is that the limit of the NN basis function is different at plus versus minus infinity on the real line. This will give you the “separates points “ property.