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> Wouldn't this methane be produced anyway as uneaten vegetation decomposes?
No, methane is only produced under certain specialised conditions, when oxygen is absent. Ruminant animals produce much more methane than non-ruminants such as poultry.
> How do NZ lambs produce less methane than Scottish ones?
I don't know if they do or not, but the total greenhouse gas emissions will be composed of methane production, which will depend on diet and breed of animal, and carbon dioxide production from the animals, the transport, and the feeds and other things. Also efficiency will improve if you end up extracting more usable meat per animal.
If you have a grass field or forest it will be carbon neutral. As it grows it will take carbon dioxide from the atmosphere which will be returned if it is burned or broken down by bacteria in the presence of oxygen, to form carbon dioxide. Methane is a much more potent greenhouse gas than carbon dioxide made from the same quantity of carbon, so you want to avoid getting into the atmosphere. However you can make a composting kind of setup that is closed to the air with anaerobic bacteria that produce methane when digesting the grass, and then burn the methane as it comes out or use it as a fuel elsewhere, and then you are carbon neutral again.