My understanding is that aerobic composting does not produce methane, but anaerobic composting does. If you're doing the composting at home you can set it up for aerobic composting. But if the waste material is going into a landfill, it's more likely to end up in an anaerobic composting process. Unless you happen to live in one of the few regions that actually have people separate out different types of waste products properly.

Right. That's why throwing plant cuttings and food scraps in the general rubbish in bad for the atmosphere. As recently as 2017 (latest EPA data) the majority of organic waste in the US was still being thrown in landfills, so there's plenty of opportunity for improvement.