Sounds like a magical fantasy computer.
How would the input and output devices work without drivers? Does "no classical software at all" mean that the drivers are somehow within the neural net?
How would such a magical neural net computer be trained? Wouldn't it require observing countless inputs and outputs for a normal computer? How would the neural net being trained distinguish between good, wanted input/output combinations versus bad, error input/output combinations?
How would you program such a computer or load new software onto it? If the whole thing is a single neural net, wouldn't the neural net need to be replaced in order to add new functionality or correct errors/bugs? How would user data be persisted while upgrading the applications?