Is there a solid proof that ensures the concept of hydrate-sourced methane reaching the atmosphere in amounts sufficient to affect the global-methane and global-CO2 budget?
I will only add some thoughts rather than a rigorous answer.
In order to answer this question one must estimate the rate of formation of hydrated methane at the surface of the earth. The agriculture waste thrown in sweet water will be hydrated and form methane. I do not follow the statistics of such process. But if one produces methane from hydration of the agricultural waste it may add an appreciable quantity to the natural generated methane as one will not completely extract the methane in the waste.