Yes, it would be un-idiomatic to say "an Australian person". English speakers would almost always simply say "an Australian". The only reason to add "person" that I can think of would be if there was some possible ambiguity otherwise.
Like, "On my ranch I have two American horses, three Argentine, and one Australian. Yesterday an Australian on my ranch broke his leg." Do you mean an Australian horse or an Australian person?
Note that in English, for some nationalities we use the same word as an adjective as for the noun for a person from that country: Australian, Canadian, German, etc. In other cases we have distinct words: British (adj)/Briton (n), French (adj)/Frenchman (n), Arabian (adj)/Arab(n), etc.
== Addendum Jan 2024 ==
"Aussie" is a slang term for "Australian". You could say, "I met an Aussie named Sheila" or whatever. That's no more or less idiomatic than "I met an Australian named Sheila", just less formal. Same considerations apply to "an Aussie person" as to "an Australian person".
Is this likely to change? I doubt it. Language changes all the time and I don't claim to be able to predict the direction. But I see no reason why this particular issue can or should change.
If the word "Jew" is offensive, that's news to me. I will refer to someone as "a Jew" without a second thought. Of course any word can be in an insult if said sneeringly or in a negative context. Like, "Oh, you program in Java, well, sniff, if that's what you want to do, I suppose I shouldn't judge."
As to "sometimes regarded as offensive nowadays" ... maybe so. Today people get offended by all sorts of ridiculous things. To some extent I try to avoid doing or saying things that others say they find offensive just to be polite, but at some point it gets tedious.