The CSIRO has the domain CSIRO.au and they're the only one site that comes to mind. All domains in Australia are required to be registered under the kind of organisation they are. There is .com.au for companies, you need a valid Australian Business Number (ABN) to register one, .org.au where you need to be a registered "organisation"[0]. Soon we'll be able to register .au domains and you can read more about the history of the domain on Wikipedia[1].
In short, the reason you never see `.au` domains is because no one can register them.
IANA delegated .au to someone at the University of Melbourne in the 80s. He devised the 2LD structure, and chose .com.au, .org.au, .net.au, and .edu.au as the major second level domains, with various eligibility requirements. Recently, auDA (the Australian domain administrator) started investigating releasing top level .au domains, but the timeline went from "late 2019" to "first half 2020" [0], and I haven't seen anything since.
Having lived in Australia, Germany and New Zealand amongst others, I find the Australian scheme the best. In Germany, everything is just .de (like in many other countries), while I find that the nz sites (.co.nz, .ac.nz ...) using a different scheme than the topdomains (.com, .edu) even worse.
