> While we're at it, doesn't the "universality theorem" (as the article calls it) basically follow immediately from the fact that the set of all continuous functions comprises a vector space?
Pretty much, but you have to show that neural networks can create a basis in that vector space which is essentially the proof presented in the article.
> If the continuous function is additive, it's linear. If it's nonlinear, you can differentiate it to obtain a linear approximation.
Differentiating to obtain a linear approximation does not give you an arbitrarily good approximation like the theorem does.
> A neural network computes linear transformations, so unless I'm missing something I'm a little surprised there's a substantive theorem for this. Is it not a corollary on the fact that we can construct a vector space of all continuous functions?
Neural networks using sigmoid transfer functions do not compute linear transformations anymore.
Importantly this theorem also states that you can approximate any function with only two hidden layers. A similar proof could not be made for a single hidden layer so it seems that the non-linearity of a single layer is not enough to form a basis for all continuous functions.