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

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.