Yes, they can after 4 years, although that was only a recent change.

The bigger factor is that Australia and NZ have free movement between the countries for those who are a citizen of either.

It's not quite that simple. A NZ citizen needs to be resident in Australia for a period of time before applying for Australian citizenship. But it is certainly a lot easier than many other countries.

https://immi.homeaffairs.gov.au/entering-and-leaving-austral...