But GP specifically says neural nets should be able to do it because they are universal approximators (of Lebesgue integratable functions).

I'm saying this is clearly a nonsense argument, because there are much simpler physical processes than cognition where the answers are not Lebesgue integratable functions, so we have no guarantee that neural networks will be able to approximate the answers.

For cognition we don't even know the problem statement, and maybe the answers are not functions over the real numbers at all, but graphs or matrices or Markov chains or what have you. Then having universal approximators of functions over the real numbers is useless.