I think a good example are calculations or counting letters: it's trivial to write turing machines doing that correctly, so you could create neural networks, that do just that. From LLM we know that they are bad at those tasks.

It also doesn't say anything about finding the desired function, rather than a different function which approximates it closely on some compact set but diverges from it outside that set. That's the trouble with extrapolation: you don't know how to compute the function you're looking for because you don't know anything about its behaviour outside of your sample.

As an example, I would like to ask how to parse balanced brackets grammar (S ::= B <EOS>; B ::= | BB | (B) | [B] | {B};) with that Turing complete recurrent network and how it will deal with precision loss for relatively short inputs.

Paper also does not address training (i.e., automatic search of the processors' equations given inputs and outputs).

Mathematically possible != actually possible.