I wish I’d never got into the public company analogy.
Look, what we have here is a constitutional monarchy under which the Queen of Australia has some apparently very important powers which, by constitutional convention, she can only exercise on the advice of her Australian ministers.
To my mind, if the Queen goes mad and tries to exercise her powers against or without the advice of the Australian ministers, she is attempting to perpetrate a coup d’etat.  We can differ about whether she would succeed, whether the courts would accept what she was doing, and whether, if they refused, they would be (a) upholding the Australian constitution and the rule of law or (b) engaging in a revolution.
None of this matters.  We could have exactly the same debate if the Queen of Australia were an Australian citizen who was Queen of no other country.  Clearly in that circumstance Australia would be independent, however deficient its constitution might be in other respects.
The question is, does Australia cease to be independent merely because the person who has gone mad (a) has British nationality, and (b) is the queen of the UK and several other countries?  To my mind the answer is “no”.  Nationality is clearly irrelevant, if only because the Australian Citizenship Act could be amended to confer Australian nationality on the King or Queen of Australia as of right, and it would be absurd to suggest that, if Australia is not now independent, that alone would make it independent.  And the fact that the Queen is also the Queen of the UK is relevant only if that is in some way connected to her unconstitutional behaviour in Australia.
If a foreign government (e.g. the UK government) were to take it upon itself to advise the Queen as to how to exercise her Australian powers, and if she were to take that advice, and if the Australian institutions were to accept that as legal and effective, Australia would cease to be independent.  But that would be equally true if an elected Australian president were to act on the advice of a foreign government.  The mere theoretical possibility that this might happen does not mean that Australia is not independent.  If it does, there is no independent country anywhere in the world.
I suppose you could argue that a “what if” scenario in which the Queen attempts to exercise Australian powers on the advice of the UK government is slightly more plausible than one in which an elected President does so;  I would accept this.  That would suggest that Australian independence is slightly more vulnerable than it would be if it were a Republic, or had a native monarchy.  But it is still independent.