Earlier quoted context omitted.
Methane’s much higher global warming potential (versus carbon dioxide), implies that there’s a threshold of methane leaks above which the warming would be worse than if you did nothing at all.
Doing the math, methane is around 30x worse than CO2. So say you pulled down 30 units of CO2 and 1 unit of Methane leaked (3%), you’re back to square one. Not to mention the Methane will be burned into CO2 again so this closed cycle isn’t so closed. This study suggests up to 9% is leaked currently: https://news.stanford.edu/2022/03/24/methane-leaks-much-wors...
At the point where you are doing enough direct air capture of CO₂ to supply a substantial fraction of the world fuel demand, you're also doing enough direct air capture to capture a substantial fraction of world CO₂ emissions. At that point you can just pump some of it into natural gas fields instead of converting it to CH₄.