Ask any government minister who they work for and the most likely answer will be something akin to 'for New Zealand' or 'on behalf of New Zealanders'.
That is how it should be. New Zealanders elect parliamentary representatives to serve in parliament. The Prime Minister appoints a cabinet (and Ministers outside of cabinet) and they take the required oath and are given the right to command the various ministries of state.
None of that will change when New Zealand has its own head of state. What will become clear is that Members of Parliament and Ministers of 'the Crown' are there to work for the people who elected them. Power and sovereignity will still be held collectively by the people of New Zealand and the right to use that power will be assigned every three years to the various New Zealand citizens who have earned the right to do so.
Today, all of the ministers appointed by the Prime Minister travelled to Government House, the seat of constitutional authority, to become members of the Executive Council. Despite the portraits hanging on the wall the whole ceremony had nothing to do with the British Monarchy. The conventions originated in the UK but New Zealand has since adapted them for its own purposes.
On paper the Executive Council advise the Governor-General. In actuality the Governor-General is there as a figurehead. The whole Executive Council is run out of the Department of Prime Minister and Cabinet where the Secretary of the Cabinet Michael Webster is also the Clerk of the Executive Council. The Head of Government looks after the running of the country. The Head of State is there to make sure power is in accordance with laws and conventions of New Zealand.
Under our proposal for transitioning the position of Governor-General into a democratically selected position (and so becoming a republic) the basic functions with respect to the Executive Council will remain. Ministers will travel to Government House and New Zealand's Head of State on behalf of the people of New Zealand will ask them to repeat the oath
“I, [name], being chosen and admitted of the Executive Council of New Zealand, swear that I will to the best of my judgement, at all times, when thereto required, freely give my counsel and advice to the Governor-General for the time being, for the good management of the affairs of New Zealand. That I will not directly nor indirectly reveal such matters as shall be debated in Council and committed to my secrecy, but that I will in all things be a true and faithful Councillor. [So help me God.]”
Athiests may not like the last part of the Oath but the issue of Oaths and how they worded is a separate (but related) issue. At some point in the transition to a republic the various oaths and affirmations will have to be amended to account for the change. Those things are the finer details and can be decided upon after the first indicator referendum when the proposed Constitutional Commission would look at the issue of what needs to change in order to bring a republic into effect. For now the core of the head of state campaign is very simple. Parliament and the Cabinet works for New Zealand and only a New Zealander can be New Zealand's head of state and represent the sovereignity of New Zealand.