The same link but with Australia and New Zealand in every chart [0] just blatantly disproves your narrative.
I'm just not that familiar with the mask mandates of NZ and AUS. AUS looks to have lifted their mask mandate on June 17th 2022? Looking at the link you sent, to my eyes, there's a clear upward trend of deaths in AUS nearly three weeks after June 17th (June 10th 2022).
The numbers for both are so low, relative to other countries like the US, so it's kind of hard to really read too much into them but NZ looks at worst to be 20% more per capita than AUS.
[0] https://ourworldindata.org/coronavirus/country/new-zealand?c...