Methane breaks down quickly, yes, but it also has a much, much more potent greenhouse effect. Taking both factors into account, releasing 1 ton of methane is estimated to be about equivalent to releasing 25 tons of CO2.
https://www.epa.gov/ghgemissions/overview-greenhouse-gases
(Carbon dioxide 82%, methane 10%)
Globally, as of 2012, methane contributes about 23% as much as CO2 to anthropogenic warming:
https://ourworldindata.org/co2-and-other-greenhouse-gas-emis...
(Carbon dioxide 35.47 billion tons CO2e, methane 8.01 billion tons CO2e).
So methane is certainly a smaller contributor toward warming than CO2 but it's more significant than you have conveyed.