The methane will eventually break down and be part of new plants, but in the meantime it will heat the planet much more than if it hadn't been there. Think of it this way: industrial scale cattle farming essentially makes and feeds biochemical reactors that turn hydrocarbons into a gaseous form that maximizes their global heating effect. Consider the counterfactual where we don't stuff billions of fart machines full of corn.

It's not exactly a closed loop, though. A lot of crops are grown using fossil fuel-derived fertilizer.

This is literally backwards - plants absorb CO2, not methane!

The obsession with CO2, specifically emitted by fossil fuel use, for me indicates geopolitical motivations (for both standpoints).