At the outset New Zealand’s free trade agreement negotiation with the United Kingdom, we were told that having a "shared head of state" would help get an agreement in place. Recent reports on the negotiations indicate that that clearly isn't the case - the British monarchy does nothing for us in terms of trade, military links or anything of any practical use, really. It has no benefits to New Zealand.
Meanwhile, it’s been revealed that the UK government is split on the recent Australian-UK FTA, which gives Australian exporters access to the UK’s market:
UK officials said Australian and New Zealand negotiators were holding firm on demands for full tariff liberalisation, which Truss was under pressure to grant in order to meet the G7 deadline, perhaps phased in over a 10-year period.
What would be beneficial to New Zealand is standing on our own two feet and accepting that we’re not part of some special club just because we have the British monarch as our head of state.