Regarding the effect of CO2 alone, a small ("trace") amount of CO2 in Earth's atmosphere does dramatically affect mean global surface temperature and does significantly contribute to establishing the present base level of mean global surface temperature, but a large increase (doubling or ten-fold) in CO2, in addition to the small present amount, does not significantly change the global surface temperature.
This is because of the physical phenomenon of optical saturation in the infrared resonance of the CO2. [See detailed physics calculation and references HERE.]
In other words, the "logic" that a small amount can't have a large effect because it is a small amount is wrong. The physics is unambiguous on this point.
Likewise, the "logic" that a large increase must have a large effect because it is a large relative change in concentration is wrong.
For example, by analogy, a small amount of a very toxic substance can kill you but doubling the lethal dose will not kill you more. That's saturation.
The quantity-of-causal-agent-to-magnitude-of-the-effect ratio depends on the causal mechanism at play, and is often not linear. Linearity cannot be assumed.