NNs connection to biology is very thin. Artificial neurons don't look or act like regular neurons at all.

I am well aware of that. Nevertheless, as a historical fact, they were inspired by real neurons, they do operate more like real neurons than do, say, SVMs or random forests, and this is the background to my original question.

If you have lots of labeled data you are more likely to use an SVM.

ImageNet is a lot of labeled data, to give one example.

As for major achievements - NNs are leading for now because ...

There is a difference between explaining, and explaining away. You seem to think you are doing the latter, while you're really just doing the former.