Just straight from Wikipedia:
A flag carrier is a transport company, such as an airline or shipping company, that, being locally registered in a given sovereign state, enjoys preferential rights or privileges accorded by the government for international operations.
By this definition, yes of course it remains a flag carrier. Legislation doesn’t matter, National ownership no longer matters either:
Today, it is any international airline with a strong connection to its home country or that represents its home country internationally, regardless of whether it is government-owned.
While Australians think of it as such, and foreigners recognise it as such, corporate or not it’s a flag carrier. I’d suggest it’s not a title for government or media to bestow, but a longer walk to mass agreement. Expensive ads help.