Computers can execute precise computations, it's just not efficient (and it's very much slow).
NNs are exactly what "computers" are good for and we've been using since their inception: doing lots of computations quickly.
"Analog neural networks" (brains) work much differently from what are "neural networks" in computing, and we have no understanding of their operation to claim they are or aren't algorithmic. But computing NNs are simply implementations of an algorithm.
Edit: upon further rereading, it seems you equate "neural networks" with brain-like operation. But brain was an inspiration for NNs, they are not an "approximation" of it.
NNs are exactly what "computers" are good for and we've been using since their inception: doing lots of computations quickly.
"Analog neural networks" (brains) work much differently from what are "neural networks" in computing, and we have no understanding of their operation to claim they are or aren't algorithmic. But computing NNs are simply implementations of an algorithm.
Edit: upon further rereading, it seems you equate "neural networks" with brain-like operation. But brain was an inspiration for NNs, they are not an "approximation" of it.