Getting it from existing fuel burning plants defeats the whole purpose.

Direct air capture is out, so it'll have to be recovered from the combustion of the synfuel. Using the Allam cycle has been explored to do this (you also have to store the oxygen from electrolysis for later use in this oxyfuel combustion cycle) but it ends up being more expensive than just burning hydrogen, if there's reasonable geology for hydrogen storage.

So, if this thermal storage scheme is cheaper than hydrogen, as it appears it will be, then these alternative synfuel schemes are ruled out.