> The only way to make a cheap synthetic fuel is from a nuclear reactor that produces heat in the 900C range (could be 700C or 1100C, but near there). You can't do that with solid fuel reactors, you need a liquid fueled reactor for that

Some of the highest temperature reactor concepts use solid fuel (see e.g. various VHTR gen4 concepts).

As an aside, some nuclear proponents claiming synthetic fuel production as some unique selling point of advanced nuclear sounds more like wishful thinking combined with admitting being unable to produce electricity at competitive price. With the 'electrotech revolution', most things will switch to being powered by electricity, leaving a relatively modest market for synthetic fuels (long range aviation and shipping, mainly, and some chemicals production), assuming regulation prevents usage of fossil fuels.

> And you need Thorium for a liquid fueled reactor.

No, why would you? You can use U235 in a non-breeding thermal reactor (Terrestrial being an example design), or you can run the U-Pu breeding cycle in a liquid fueled fast reactor (such designs use chloride salts as the fuel carrier rather than FLiBe).

> That's why this design is so popular.

So popular that despite being invented in the 1960'ies, it hasn't yet progressed beyond the prototype stage?