We do not have any information if New Zealand will halt flights from the United States if there are Covid cases in the future. The situation can change very quickly and it will depend on how the Ardern administration chooses to handle the situation.
While over 40% of the US population (June 2021) is fully vaccinated, in New Zealand only 4% of the population have been fully vaccinated to date (about 200k people). This is a result of lack of access to vaccinations as well as lack of need- while the border is closed there is simply less of a need to race to vaccinate the population.
Our hope is that the country provides access to vaccinations to all who want them by the end of 2021. The US, at the height of its vaccination rollout, administered as many vaccinations as the entire population of New Zealand in a day. Currently (June 2021) there are about 1.5 million vaccinations administered every day in the US as the number tapers down. With a population of only 5 million we are hopeful that New Zealand will be able to quickly disseminate shots when they are available.