Not so much - it's not really accurate to talk about a single approach given there's nine different governments here. New Zealand got a lot of press internationally for doing a quick lockdown back in March, but I think that's largely because it's a white country and so was focused on over other countries because of racism. And, being a unitary state, it is easier for people overseas to understand compared with a federation with different governments doing somewhat different things.
Even then, "laissez-faire, live-with-the-virus" was never really the approach anywhere, especially from the states who control every lever except international borders and welfare funding and very quickly got the federal government (possibly reluctantly) to play ball on the welfare front (with regrettable exceptions). There's a bit of diversity between the states in how they approach things, but it largely comes down to instant lockdowns on the identification of first case, vs test/trace/isolate as the primary method of controlling each cluster which leads to things going naturally back to zero.
Generally speaking, if people have used the phrase "live with the virus" it has not meant uncontrolled spread but rather meant maintaining some level of restrictions and record keeping measures to aid test/trace/isolate, on the understanding that no quarantine is foolproof and you're going to see clusters emerge sometimes. It's a criticism of the false certainty of relying on borders and quarantine alone.
The timeline is roughly:
Border closures from late March meant that there was a fairly limited spread in the first wave with most people being isolated and not passing it on to many people. That mistakenly disembarked cruise ship only led to a limited number of secondary infections for instance, since everyone was contact traced pretty quickly. Global travel bans and mandatory quarantine for a limited number of international arrivals was instituted in March. Lockdowns were instituted over most of April and they killed off the existing spread - by mid May the thing was largely eliminated and under control by contact tracing. There were single digit national daily cases through much of late May, and actual zero days by June. Then the Victorian second wave happened, which wasn't an experience replicated anywhere else in the country due to either having no virus or successfully finding and tracing it.
So pretty much since July the situation has been:
- NSW successfully and repeatedly suppressing clusters and repeatedly returning to zero cases without city-wide or state-wide lockdowns. Clusters usually taking about 3 weeks to play out even with big seed events.
- Victoria sat in a four month hard lockdown from July to October, the rest of the country closing its borders, while Victoria rebuilt its public health systems to be able to do what now looks like a perfectly decent test/trace/isolate system
- The rest of the country largely avoided new spread by the simple expedient of NSW handling the majority of international arrivals, meaning fairly limited quarantine numbers in other states, which have each only rarely produced any leaks of their own.
One interesting inversion that's actually happened recently is that New Zealand has shifted way from instant lockdowns at the first sign of a community case, towards a more New South Wales-style reliance on test/trace/isolate with localised lockdowns reserved for more serious seeding events. So their recent Northland case, they didn't react the same way they did a few months ago.
By contrast, some Australian states (Queensland, South Australia, Western Australia) have recently started doing those instant lockdowns at the first sign of an infected quarantine worker, although as it turned out the SA one was based on false information and rescinded the day they realised this. And the Queensland one was about about fear of the new strains, which I suspect will have subsided a bit after there were no transmissions from the case there.