Would Australian government be able to function if Australians were not under the reign of royalty from the UK?

China and India are the new parents of Australia.

The Head of State in Australia is the Queen of Australia. When she is not here, she is represented by an Australian Governor General. Government is controlled by the Parliament and the Queen or her representative act only on the instructions of the Prime Minister - the same way if works in the UK. The Queen is largely irrelevant to the functioning of the government in Australia.

Yes. On the other hand it seems these days the UK government can't seem to function without the supervision of Brussels given how they're squandering their Brexit opportunity.

Unlikely. Aussie politicians have always needed oversight from poms - whether it be from the royals themselves or from the royals’ viceregal governors they send down to oversee the running of Australia. Since its founding, up until this very day, Australia has been supervised by the monarchy under royals from Britain.