If you're doing carbon sequestration you skip the part where you convert it to methane and just sequester the captured CO2.
Given the scale of the problem, something like that will (should?) be done. If we take it seriously and figure out a political solution, eventually every current natural gas well will be converted to CCS and run in "reverse", as it were.
However, the energetics are such that most carbon capture, barring nanotech magic, will be done via enhanced weathering or tree burial.
If they make it profitable to get carbon out of the air which this suggests it will get a lot of private investments. This will also make carbon worth something so instead of letting it out in the atmosphere there will be effort to capture and sell it to the solar methane generators.
GP asked about sequestration. Most of the productive economic uses for CH4 require burning it. Liquid synfuels are great and wonderful but they are carbon neutral!
You can use atmospheric CO2 as feedstock for plastic production, but world plastic production is only 380 million tonnes a year, while there's more than 1 trillion tonnes of excess CO2 in the atmosphere. It would take two and a half millennia to convert that to plastic.
Given the scale of the problem, something like that will (should?) be done. If we take it seriously and figure out a political solution, eventually every current natural gas well will be converted to CCS and run in "reverse", as it were.
However, the energetics are such that most carbon capture, barring nanotech magic, will be done via enhanced weathering or tree burial.