Exactly. It's not "making fuel" in the sense that would be energy positive. It's using CO₂ as feedstock into an energy conversion process that transforms energy otherwise obtained into a form usable in jet engines.

I hope these misleading "turned into" style of headlines go away when energy input is required. Carbon + Hydrogen + Oxygen + energy + information can be "turned into" pretty much any organic substance.

We already have that in the form of fission. The problem is our societal lack of interest in fission.

Even if we had access to free electricity, wouldn't it still be cheaper to utilize existing petroleum infrastructure?

I mean, until the reserves are depleted the energy is basically free, we just need to pump it up and refine it.

(Then again, solar power is basically "free", we just need to catch it and store it... so my 2nd paragraph is maybe kind of dumb.)

The sibling comment from jeffbee is essentially correct, depending on which part of the "petroleum infrastructure" you're referring to in your comment: the price of extracting and burning fossil hydrocarbon fuels does not reflect the cost of their externalities, primarily climate change driven by CO2 emissions.

However, you are right in the sense that taking advantage of our current hydrocarbon distribution infrastructure would save capital costs versus switching to, e.g., an all-hydrogen or all-electric energy infrastructure. The trick is to get those hydrocarbons from non-fossil sources, which is a solved problem from a technical perspective but very much still unsolved from an economic perspective.

Not when you count the external cost of taking it out of the ground and putting it into the atmosphere.

If you had the renewable energy surplus necessary to do it, you'd make as much liquid hydrocarbon as you could, and pump it back into the ground.

The German government recently updated its estimate for the societal costs of putting a ton of CO2 into the atmosphere to 195 euros (up from 180 euros a few years ago).

If you pay the energy price you can upgrade it to almost anything you want. But that is the question, how many joules of energy do I need (end-to-end) to spend to store one joule in a synthesized hydrocarbon.

Is the heat given off by burning fossil fuels significant compared to the amount of heat trapped by the released CO2 over it's lifetime in the atmosphere?

... or when wind / solar drops another 40-50% in cost and we have excess capacity.