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It's one of those things that's fine technically but economics is a problem.
Plenty have tried to produce synthetic fuel of some sort over the past century and ultimately most failed not with doing it as such but on the economics.
Germany did it during WW2 for obvious reasons using coal as the feedstock and South Africa built some facilities during the sanctions era and still runs them producing a reported 150,000 barrels per day from coal.
There's nominally 233,700 barrels per day of gas to liquid fuel capacity worldwide, three quarters of that in Qatar and the rest in South Africa and Malaysia. For coal to liquids nominal capacity is about 150,000 barrels per day in South Africa.
Historically the world's first two stage gas to petrol plant operated in New Zealand from 1985 to 1997. Nominal capacity was 14,450 barrels per day of 95 RON petrol.
Historically there's also been a very large number of attempts to produce liquid petroleum from oil shale (not to be confused with shale oil being extracted in the US and elsewhere which is a very different thing). Locally some was produced successfully in Qld, NSW and Tas indeed the latter even has a specific type of shale named after it (Tasmanite).
Ultimately though it's a problematic process economically and today the only real production of liquid petroleum from oil shale occurs in China (reported to be about 82,000 barrels per day) and Estonia (about 4500 bpd). There's also historically been production of gas from oil shale in various countries, most notably in Estonia. Of all oil shale used, most was simply burned as pulverised solid in a coal-like manner to generate electricity with no attempt made to produce liquids or gas.
Now consider the world uses over 100 million barrels per day of oil and that the sum total of the synthetic fuel plants, all based on fossil feedstocks, is less than 0.5% of that. Attempts to produce liquid fuels from something other than oil pumped from the ground have really only succeeded at all in countries with low labour costs, minimal regulation and/or outright subsidies and even there they haven't grown beyond a very limited scale. And that's with using fossil feedstocks available locally at well below world market prices and without any regard to CO2.
Hence my personal view, although hard to prove, is a pessimistic one. If it's so hard and expensive to make jet fuel, diesel etc economically when starting with cheap or even free coal, gas or shale rock, which are already hydrocarbons, then it's going to be at least one and perhaps two orders of magnitude harder and more expensive to do it using CO2 extracted from an extremely dilute source combined with hydrogen from electrolysis.
Biofuels are different, in that the economics are a lot better, but they are extremely scale limited. Well, they are unless we're going to outright trash the natural environment to obtain sufficient biomass.
There's a proposed plant in Tasmania that if built will produce sufficient methanol that, if converted in a second stage process to hydrocarbon fuels, yields about 6PJ per annum of liquid fuel.
To put that into perspective, Australia uses about 325PJ per annum of aviation fuel alone and it uses 2190PJ of liquid fuels for direct uses in total (in layman's terms that means all uses except electricity generation).
So this plant in Tasmania will replace about 0.27% of Australia's oil use and to do that it needs, wait for it.....
500,000 tonnes of green wood (or 300,000 tonnes if it's dry wood) per annum as the carbon source.
Plus 240MW of electricity constantly to produce the hydrogen.
There's a very real limit on how far this can be scaled up before it runs into major resource constraints and environmental objections with that biomass consumption.
Even just scaling it up to replace aviation fuel, nothing else, needs over a million tonnes of wood each and every fortnight. Obviously all that doesn't have to be done in Tasmania, but even spread nationally over a million tonnes of wood per fortnight, every fortnight forever, is a lot of trees.
I'm not against it as such, and I'm not suggesting it won't be done to some extent, but there's a foreseeable scale problem here. It works to the extent there's waste wood or at least wood that nobody objects to being cut but there's a scale limit to that.
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