In my case, I travel a lot for business and my elite profile with the airline has all my info including about PP and GC. I can get an online boarding pass and go straight and board the aircraft if I am traveling to my home country as I am holding the passport. But if I need to connect to an other country where I need visa then airline staff verify my PP and GC before they give the boarding pass. In the example here , even though OP holds Australian PP, he must be connecting somewhere ( as there is no non-stop flight ) so he needs to stop at the airline counter and they will ask GC. Again this is from my experience of about 3 to 4 international trips a year.
Regarding the bag check in questions, you have to swipe the GC on the machine ( at least in my case) and the baggage tags are printed behind the counter and the airline employee place the tag on the check in bag and they usually verify the travel document before they do.
As I mention in my first post, I am not familiar with the Australian law but it is Illegal where I came from to enter the country using one PP and leaving with another and I do have first hand knowledge of people went to jail . Unlike in the US most countries has an immigration check point before you leave ( by Air) and they will verify your entry into the country by usually looking at an entry stamp on the travel document ( PP) and if they can’t find it you need to explain the reason. This may be ok in Australia but one need to check. That is why I asked the question whether Dual citizenship in Australia is automatic, meaning after you receive the citizenship in the US the Australian citizenship and PP is valid by default or you need to file some paperwork. I know some countries require their citizen to file some paper work after they receive citizenship from an other country. Excuse my long post I got into a fair amount of trouble traveling internationally, I didn’t do anything illegal but you get into some interesting situations sometimes
