Where do you see the aboriginal flag flown? I am aware of a few places, but the place I see it most is online - where it is being used to show that an entity is based in the aboriginal community and/or supports the aboriginal community.
So you want the Australian government to expropriate private property without compensation? There’s a clause in the constitution that specifically forbids that, for good reason.
In hindsight, maybe there should have been a competition to choose the Aboriginal Flag, and the winner would cede rights into the public domain in exchange for a prize (monetary, presumably). That didn’t happen, and wishing it did happen will not change the situation. The owner of the intellectual property is well within his rights to licence that property; while taxpayers or a philanthropist could have stepped forward to place the flag into the public domain (or limited public domain to prevent misuse), for the moment those licences are controlled by commercial operations that presumably offered more to the IP holder than any other bidders.
I am not arguing the merits of whether the rights should not be in private ownership; I am merely pointing out that since that is not the case there are only a few ways to change it - and you have already ruled out the obvious one, of taxpayers paying for those rights.