No it is not closed, at least not like the cycle you mentioned.
There is a carbon cycle for leaves, carbon in the ground, and CO2 in the air. Plants use CO2 from the air to make leaves/biomass. Plants die and the biomass is decomposed (or burned etc) and the carbon goes back in the air (as CO2).
But there is no such cycle process for methane. Cows put methane in the air, but plants don’t remove methane from the air. Nothing on the ground removes methane from the air. It just floats around, slowly broken down by solar radiation in the atmosphere but that’s it.
Also methane has 80x the warming power of CO2 so it’s worse in that way too.
Source: I took a university course on atmospheric chemistry. It’s VERY complicated.
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