Energy cycle centered around methane production / consumption
Vehicles that run on natural gas already exist, and have done for many years - you can find CNG (compressed natural gas) cars and larger vehicles around the world.

But there's very little synthesising of methane because there are very few places with a sensible carbon price, so it's just cheaper to extract natural gas and burn it.

Research is ongoing on methane synthesis - this is usually done under the name "power to gas", which can refer to hydrogen or methane. Either way, it usually starts with the splitting of water to get hydrogen, and then is sometimes combined with methanation of the hydrogen. The latter is an exothermic reaction, so combining the two processes can give pretty good efficiencies (see e.g. this paper).

Hydrogen / methane for cars is looking like a lost battle: electric cars have almost certainly already won; it's just going to take a few years until they completely dominate. However, there is still the possibility that larger vehicles (heavy goods vehicles, boats, aircraft) might settle on hydrogen or renewable methane as their decarbonisation option.

By the way, just because a fuel-cell might use rare-earth metals, doesn't make it unsuitable for mass market. Most rare-earth metals aren't rare, despite the name. Fuel cells have lots of issues, and fuel-cell vehicles really haven't achieved much yet, but that's to do with lack of reliability and longevity, rather than scarcity of components.