The “CO2 saturation effect” is a fallacy based on misunderstanding of how heat transfer works in the atmosphere. Heat passes from the earth’s surface by convection up to an altitude from which it can radiate. Increase the CO2 in the atmosphere and you increase that altitude. That increases the adiabatic cooling in the convection, meaning a higher surface temperature to maintain the same temperature at the radiation altitude. The earth’s surface will continue to get warmer as we continue to dump more CO2 into the atmosphere.
See the Wikipedia article on Climate Change. It has two important graphics - the first of global surface temperature since 0 AD. The temperature rise over the last century dwarfs all the previous effects including the Medieval warm period and the following “little ice age”. The second is CO2 percentage in the atmosphere over the last 800 years. It’s fairly consistent until the Industrial Revolution takes it off the chart. You can only miss the correlation if you’re in willful denial.