> It becomes CO2 (a relatively minor greenhouse gas per unit weight, but long-lived)

Although it should be noted that 1 tonne methane decays into around 2.5 tonnes of CO2 (I don't remember the exact number but it's around that) and GWP is measured by weight.

So even after it's decayed, methane has a higher GWP than CO2. Which is why its GWP remains much higher than CO2 even over extremely long periods: methane has a GWP of ~80 over 20 years, ~30 over 100 years, but is still around 7 over 500 years, despite a lifetime of only 12 years.