Although your facts are correct, the reason CO2 causes warming, and the reason it doesn't saturate too, are more complicated.
These complications are the reason why it wasn't until the late 1940s and the advent of high altitude aircraft that these areas of confusion weren't definitively settled.
A better model (one that behaves more like the real world) is to consider the Earth as a black body where the surface is a mile or two up in the atmosphere rather than on the ground.
CO2 (plus water vapour) are what control how high into the atmosphere that surface is.
Because of the lapse rate, the ground will be warmer than the surface of the imaginary black body.
As CO2 increases, the height of that black body surface increases therefore it's temperature decreases. However, if the temperature decreases, the amount of radiation escaping to space decreases while the amount arriving from the sun stays the same, so the ground starts to warm up.
Eventually, the ground warms enough that the black body surface is hot enough to now be in equilibrium with the energy arriving from the sun.
And because that surface raises if CO2 is added regardless of how much CO2 there already is there is no "saturation" point where more CO2 doesn't cause warming.