You mean liquefied natural gas? Yeah, that's doable, but as Europe is finding out right now, it's very expensive. You have to keep the liquefied gas sufficiently chilled all through transport. You wouldn't have to do that with methanol.
The vapor pressure of methane is about 6.25 MPa. For comparison a bike tire has about 0.4 MPa, a car tire closer to 0.2-0.25. So you need a pretty damn good compressor and pressure vessel / pipeline, versus a simple tank for methanol. Also worth noting that the methanol can later be easily converted to dimethyl ether and used as a cleaner-burning diesel fuel (vapor pressure ~0.5 MPa), where it may be possible to convert existing engines:
Above its critical point, (-82.59°C) it won't condense into a liquid. It will continue compressing into a supercritical fluid, which will take a very high pressure to equal the density of liquid methane. The gas is about 0.7g/L at STP, the liquid is 422g/L at 1 atm at its boiling point. You'd need 600 atmospheres or 8862 PSI. A typical cylinder is around 2500-3000 psi.
I haven't read TFA yet (edit: as I thought, it's very light in details), but I know it's a hassle to spring up gas distribution logistics where it's only a secondary product (oil wells). The question is: is this new method simple enough to put in practice at production sites?
Liquified Natural Gas (LNG) is mostly methane with some ethane and is typically transported in cryogenic containers as it would require too much pressure to otherwise keep it in a liquid state. Even when transported in cryogenic tanks there is some boil off that is either captured and recompressed or used as fuel for the tanker.
Liqified Petroleum Gas (LPG) is a mixture of propane and butane which does not require cryogenic storage and will remain liquid.
Liquid gas is compressed methane, and some industries and cars already run by burning it. So the infrastructure already exists.