The most damaging greenhouse gas produced by livestock is not carbon dioxide, it is methane. The whole "cows farting" problem you hear about is related to methane. Ruminant animals (sheep, deer, cows, goats, etc) has a specific way to digest grass, which produces a lot of methane which is let out by farting.
I had a quick look and have not seen any articles saying birds goes through the same digestive process and produces the same amount of methane as ruminants. I am reasonably confident that only ruminants produce (at least a bad level of) methane.
So even if NZ had as many moas as we do now game runimant, there would not have been as much methane produced and not as much green house effect.
Another big methane producer, surprisingly, is rice paddy. In other words, consuming rice, sadly, encourages methane producing agricultural activity.
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