[jellicle]

You are mostly right, but the missing part is that you don't need to turn the CO2 back into a hydrocarbon fuel, you just need to turn it into something that isn't gaseous CO2.

So carbon capture hinges on the idea that we can find a low-energy route that involves a chemical reaction with CO2 that produces something that isn't a fuel but isn't gaseous CO2 either. And that we can find a LOT of it.