I suspect the point is that we would be better off just burning the methane at the source and putting the energy we would have used converting it to methanol instead towards whatever energy the methanol would have eventually been used for.

Spending $1 today to get $0.50 tomorrow isn't a good idea.

It really depends. Methanol is liquid at STP conditions, whereas methane isn't, and either needs to be stored under pressure or cryogenically. Methane also boils off, leaks, and is a potent greenhouse gas. Methanol breaks down naturally in the environment, etc.

Most appeals to the second law of thermodynamics are meaningless without additional context. We routinely make energy and chemical conversions that lose recoverable energy, but we do so because either we need the feedstock for something, or because things like fungibility and storage matter.

I mean let's be candid, the folks working on this have likely forgotten more about thermodynamics than most folks here have ever learnt. Methane to methanol has been a holy grail of chemical engineering for a while, and I'd imagine there's a lot more to it than first year physics.

I'm saying this will only become viable when the energy being put into the conversion to methanol (not creation of methane), is lower than the energy stored in the methanol. That does not violate thermodynamics.