ANSWER:
The controversial events of 1975 had nothing to do with the monarchy whatsoever. Sir John Kerr merely exercised his power as vested in him as Governor-General by the constitution - namely section 64 which gives the power to the Governor-General to '... appoint officers to administer such departments of State of the Commonwealth' and that 'Such officers shall hold office during the pleasure of the Governor-General' (in other words the right to hire and fire ministers and thus in effect governments). In fact Sir John did not even inform Her Majesty of his actions until after he had dismissed the Whitlam Government.
Regardless of whether you agree with a Head of State having the power to sack governments the fact is a republic in itself would not stop it from happening again. Merely rubbing out 'Governor-General' and replacing it with 'President' would just give all the current powers of the Governor-General to the President. Even if becoming a republic in itself would remove the power to dismiss a government, the Turnbull/Keating 'Bipartisan' republican model, which we are to vote on in the forth coming referendum, expressly states that the powers currently exercised by the Governor-General shall be exercisable by the President! In other words, becoming a republic would only change how the Head of State is appointed not the powers themselves.
The only time the Queen has ever been asked to become involved in Australian politics was when the Speaker of the House of Representatives on behalf of the Labor Party wrote to the Queen asking her to overrule Sir John's decision and restore the Whitlam Government. The Queen, through her Private Secretary, pointed out that under the constitution she did not possess the power to do so.