They do correlate pretty strongly, but the temporal lag indicates that it's actual rising temperatures that increase CO2 levels, rather than the opposite. That's likely because warmer water retains less dissolved CO2.
Solar activity, ‘milankovitch’ cycles have a big effect on temperatures. I’ve heard Solar scientists call out the nonsense that CO2 is driving temperatures. CO2 change is a down stream effect of temperature change as you say.
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There's also some newer work on latent heat in the Earth's mantel and core. There are geophysical cycles where more mantel is melting than solidifying, which releases latent heat to the surface.
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That sounds plausible to me
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