Up until Federation it was pretty easy to jump ship and live in the Australian colonies. A small white population only grew because of it, and easy emancipation of convicts.
It was concern over Chinese migration and the importation of enslaved Pacific Islanders that led to migration control in the early 20th century and the White Australia policy. A prospective migrant could be given a dictation test in any European language at the Government’s discretion. You fail, you don’t get to stay.
Post World War II, Australia kept importing Brits on assisted passage (known as 10 pound Poms) but also opened the door to European refugees, displaced persons and migrants. Gradually the non-white numbers started to rise, partly as Australian foreign policy and national security pivoted away from reliance upon the UK to multilateral relations with the Asia-Pacific and reliance upon the US. The White Australia policy was finally fully swept away in the 1970s, just in time for waves of based Vietnamese boat people and crazy Lebanese.
The current Migration Act was passed in 1958. It introduced the system of making migrants prove they have something to contribute. God knows the system is flawed, but the numbers coming in are basically set by the Government of the day, and it manages visas. It’s not that Governments haven’t caught up (the Act is amended very frequently), it’s that the current Labor Government relies on migrant voters and doesn’t want to look too racist. Before anyone starts with blaming (((them))), Jews vote conservative here. They’re not wild about importing Jew-hating Muslims who shoot up their parties.
Your coworker is right in one respect: cheap air travel certainly had an impact, but not because it made it easier to get here. It made it easier to visit your home country once you were here. Older migrants knew they were pretty well stuck here and assimilated. The current lot fly home frequently, import half the village, and don’t think learning English is necessary. Australians are pretty laidback, but the rise of the anti-migration party One Nation shows the Government is failing. Ironically it’s electorally good for the Labor Government as it splits the conservative vote.