And like you, I have wondered why a temperature lagging trace gas is considered a driver of temperature variation. If the science is settled then where are the mathematical proofs? Where is the statistical analysis? If they are referring to the software package you can download from ipcc then that doesn't contain proofs.
It lags because it is an amplifier of other warming (e.g. solar). For example the ice ages start with some cooling from lower solar (and other solar effects that change with the earth's tilt). Once that cooling kicks in, then CO2 starts to get absorbed in the ocean to cause more cooling. The same works in reverse.
Since the end of the Little Ice Age (mostly a solar effect), the earth has warmed naturally which would have produced about 5-10 ppm more CO2, that is we would have gone from 280ppm to about 290. Instead we have bumped up to 390. So nature has gone into reverse and absorbs about 1/2 of what we put out each year. The actual amount absorbed is actually much greater because nature still releases as well. If we stopped producing CO2 today, levels would return half way to equilibrium in less than 40 years (the equilibrium now being a little higher than the 280 starting point, perhaps 300).