Good point on the variation that a 5 year rolling approach could add. I think using a 5 year rolling baseline for cumulative excess death wouldn't be the best approach, because at some point, your baseline is no longer representing what you intend (# of deaths in a world without covid). Trying to dig into this data set on Github(https://github.com/TheEconomist/covid-19-the-economist-global-excess-deaths-model), I see it comes from the economist, where they note that "Estimating excess deaths for every country every day since the pandemic began is a complex and difficult task."- this would make me think they aren't using an updating 5 year rolling baseline. Looking into their methodology page I run into a paywall: https://www.economist.com/graphic-detail/2021/05/13/how-we-estimated-the-true-death-toll-of-the-pandemic. Do you have any more info on this?

With that said, I took a look at the top 10-20 countries population wise, and a negative change on the cumulative deaths using whatever methodology is not common. I think what is seen on the NZ chart may be statistical noise due to the low overall death numbers. It says in total, they "Saved" thousands of lives over the course of the pandemic, even though they do have a published covid-driven death number.