Are neural networks inherently imperfect because they mimic an imperfect machine (human)?
It's better to think of neural networks as inspired by human biology, not that they're using exactly the same mechanism.

The notion of "inherently imperfect" isn't well-defined, so the question isn't meaningful.  And, no, neural networks aren't always imperfect.  For instance, suppose the learning task was "label everything 0".  If you give a neural network a training set with one million examples, all of which have the label 0, then the neural network might well learn to output label 0 on all possible inputs.

As far as moral implications, it's not clear that it matters much why neural networks are imperfect; what's important is that they are often imperfect.  But moral implications are probably more a matter of the study of ethics and philosophy than the science of computing, so I'll leave that there.

Finally, I encourage you to avoid binary thinking.  Don't divide the world into "perfect" vs "imperfect".  An engineer would say that in practice nothing is perfect; it's all about degrees of imperfection.  One system might be good enough, or better than the reasonably available alternatives, even if it's not 100% perfect.