Does POS pollute more if POW can be used to mitigate landfill methane?

My understanding is that bitcoin, because it uses the proof of work (POW) consensus algorithm, emits a considerable amount of greenhouse gas. In contrast, ethereum now uses proof of stake (POS) and emits considerably less.

However, according to Daniel Batten, bitcoin is good for the environment because it mitigates landfill methane. According to this argument, landfills release methane which are significantly worse for the environment than carbon dioxide. Bitcoin provides landfill companies a profit incentive to use the methane from landfills to mine bitcoin. This causes combustion of the methane, which converts it into carbon dioxide, which is a less potent greenhouse gas. Bitcoin then acts like a carbon offset. If you buy carbon offsets, you are effectively paying someone to reduce their greenhouse gas emissions. By buying bitcoin you are giving landfills a profit motive to convert methane into carbon dioxide, which decelerates climate change.

If this is true, it means that ethereum's transition from POW to POS is bad for the environment because more crypto market cap has now moved from POW crypto to POS crypto, so there is less incentive for landfills to convert methane into carbon dioxide.

There are some arguments against this e.g. why doesn't a landfill simply harvest methane emissions and instead direct the electricity into the grid rather than into bitcoin miners? The counterargument to this is that the payoff from feeding this electricity into the grid may not be greater than the payoff from mining bitcoin, which explains why currently most landfills do not do this and instead release methane into the atmosphere or try to flare the methane.