The fuel need not be 10x as expensive if you mine at the natural sources of hot CO₂. Which are steel-making furnaces, cement-making furnaces, and, well, carbon-burning power plants. All these industries can provide you with plenty of CO₂, reliably, at a zero or negative cost.
This makes the synthesis much more efficient, because you need far less energy and space to capture the CO₂.
Even if all power plants could turn carbon-free, steelmaking and production of cement cannot, they involve CO₂ as a key chemical step. Until 100% of steel is recycled, and concrete is replaced entirely by something else, you will still have stable, rich sources to run your synthesis off of.
Scales poorly, and these C02 feeders can ruin your business overnight by imposing or jerking up fees once your operation is even remotely viable. What are you going to do? Teleport your facility?
The location advantage then becomes your greatest disadvantage.
it's just like installing solar in parts of the Sahara. Land is cheap and sunshine is abundant. Until an Al-Qaeda affiliate seizes one of your solar farms or the local government is overthrown and the coupists are trying to extort you. Now, how much will you spend hiring mercenaries to retake and occupy a foreign country, even if you discount international backlash?
Scales poorly — any oil well has a particular debit, naturally limited. So will this.
The conflict of interest is real. This is why I expect the same companies that produce the CO₂ to process it into fuel. Say, steelmakers need a lot of pure oxygen, this is why they sell the liquid nitrogen they produce along the way. Similarly they extract and sell a number of metals that occur in the input ores in low concentrations and are not worth mining by themselves.
Regarding Sahara, the things are sadly as you described. But large industries are usually in politically more stable areas.
This makes the synthesis much more efficient, because you need far less energy and space to capture the CO₂.
Even if all power plants could turn carbon-free, steelmaking and production of cement cannot, they involve CO₂ as a key chemical step. Until 100% of steel is recycled, and concrete is replaced entirely by something else, you will still have stable, rich sources to run your synthesis off of.