The problem is the cost of getting the carbon and hydrogen to make the natural gas. If you're making a synthetic fuel, you might as well make methanol, which can be stored as a liquid, and is about as easy to make from syngas as methane.

Yeah, did mention the methanol option as an additional step once the natural gas is created. But is the cost of getting the carbon and hydrogen that high? In a P2G system you'd be electrolysing water (plentiful) and combining it with carbon dioxide (also relatively plentiful, although it can also be linked to an existing fuel burning plant for better efficiency). There are existing production systems using this approach right now with surplus energy from renewables described in the wikipedia article.

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.

It's more efficient if you plug it into an existing plant which is burning fossil fuels anyway (maybe even an existing natural gas plant, which if the loop was closed would basically make it a really stable chemical battery, which isn't really defeating the purpose at all!).

But, if you read the wikipedia article, you can see there are prototype plants using ambient air capture. It's probably a bit less efficient, but since it's actually reducing carbon levels, it's even better than just a battery.