Even theoretically, no they can't. They can theoretically model any continuos function.
Plus, even for continuous functions, the theorem only proves that, for any function, there exists some NN that approximates it to arbitrary precision. It is not known whether there is some base NN + finite training set that could be used to arrive at that target NN using some algorithm in a finite number of steps.
For example, for a function f(x) defined on R with f(x) = -x if x < 0 and f(x) = 7+x if x >= 0, how would you approximate it by a continuous function g(x) with precision lower than, say, 1 (i.e. |f(x) - g(x)| < 1 for any x in R)?
And of course, there are functions with much worse discontinuities than this.
Of course you then need to compensate with residuals, initialisation, normalisation, and all that, but it’s a small price to pay for scaling much much better with compute.
Yes that's exactly my point. A lookup table is a universal approximator. Good luck making AI with LUTs.
It's kind of like the halting problem or the no-free-lunch theorem. Interesting academic properties but they don't really have any practical significance and often confuse people into thinking that things like formal verification and lossless compression are impossible.