Nuclear physicists and astronomers have worked out how much of the various heavy elements should be present in space, as a result of synthesis is stars and ejection in supernovae. There calculation shows that it pretty much accounts for all of the carbon that we find.
The idea is that the carbon was drifting around in space, and is part of the mass that accumulated to form the earth and other bodies.
You are right, that what happens in the stars is not important as part of the carbon cycle, except as it affects carbon 14. The carbon cycle itself does depend on living organisms. But the carbon was mostly on earth before the first life form, so where the carbon comes from is not important when investigating the carbon cycle.
It is correct that the nuclear production of carbon in stars is not of current importance in the carbon cycle. But then the source of carbon is not important either. We can just take the carbon for granted as always present on earth. That's why your IC argument over carbon seemed so out of place.
You can, if you wish, marvel at carbon and its properties. You can marvel at water, and its peculiar behavior when freezing. These, and other facts, are important to life as we know it. But, in so marvelling, you would be approaching something more like the fine tuning argument than the ID argument. The fine tuning argument is not opposed to evolution, but it is opposed to atheism.