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.
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.