Well we should be moving on from CO2 because in this area the effect is nearly saturated. Doubling CO2 from it's current level won't have nearly the impact the previous doubling had.
It's not the saturation at the peak absorbtion frequencies that is driving the warming, it's the widening of the curves which raises absorbtion levels in nearby frequencies. The "saturated" argument was one of the first objections when the science was poorly understood. Krauss's book Physics of Climate CHange has a good account of this history. Check IPCC summaries to see how many models from the literature predict flattening at 0.7C additional warming if we don't curb emissions (spoiler alert: slim to none with many positing 5C or more additional warming in the 2100's if emissions aren't curbed).
CO2 levels are current 1.50 times pre-industrial levels, so "previous doubling" is misleading.
Right but it's not going to do much of anything temp wise from here, maybe .7C higher at worst. This is a useful tool to see this: https://climatemodels.uchicago.edu/modtran/