Earlier quoted context omitted.
That's a misunderstanding. The methane stays in the atmosphere a few years until it reacts with oxygen to CO2. So first you have the multiple times worse effect of CH4 and then you're left with CO2, which is still a greenhouse gas. You're left with CO2 either way, even when you burn the methane. That's why it's so bad. And that's why actually when companies or politicians tell you about "clean" gas, it's very mislead…
Sure, but the question is still whether that carbon came from a fossil source or was part of the cycle to begin with. Let's say a cow eats grass, produces methane, the methane gets converted into CO2, the grass grows back. No net increase in CO2. If, on the other hand, the methane came from natural gas, you have permanently added carbon to the cycle.
I think we're only able see some carbon sources as "outside the cycle" due to our infinitesimal lifespans.