Assuming it's possible to use this process on a home scale with a decent solar grid and energy storage, it seems to me that you would be sequestering the carbon only very temporarily. Wouldn't the electrical energy used to produce the methane heat your home more efficiently (assuming you have an electric heating system) as you would be inviting energy losses through the Sabatier process?
Seems you would need to produce industrial levels of methane and distribute it (which would almost certainly be carbon expensive in it's own right) as substitute for naturally occuring methane being extracted from deposits.
It's a cool idea though. If there were a process to produce solid carbon, this would be much more promising (disclaimer: definitely not a chemist).
That's not as satisfying as just plain sequestering the carbon somehow. I mean, it would almost certainly be better to just run an extension cord to a neighbors house and let them run an electric space heater or heat pump off of the excess electricity.
Maybe methane could be used as an ingredient to make something else that's possible to store or bury long term, like some kind of plastic or liquid? (Most of what I know about the processing of hydrocarbons is probably wildly inaccurate because I learned it from Factorio.)
Probably (over) insulating the house is the most efficient way of minimizing energy loss. But then does an over-insulated house get overly warm in summer or easier to cool?
The air-sourced methane could be used for things that are hard to electrify/de-carbonify, and could reduce the use of ground based sources.
I'm wondering at what level of carbon tax it makes more sense to extract the carbon from the air to turn it into plastic than to use ground-based oil.
Methane is a worse greenhouse gas than CO2, but if you burn it as fuel it's a carbon-neutral energy source.
Alternatively, you could use electrolysis to make hydrogen, but hydrogen is harder to store.
(I'm not a chemist, so I have no idea if this could work. And it probably wouldn't be very practical even if it did.)
I do like your idea of experimenting with a rock crusher. The article mentions that finding the least-energy means of crushing rocks to the appropriate size is something of an open problem.