Longer trips and heavy duty applications like shipping would be the main benefit. As far as efficiency, burning it in a large scale power plant to make electricity to power an EV is more efficient than burning it to directly power a vehicle. But certainly there are applications that make sense.

Keep in mind though methane capture from waste can only scale so much and trying to produce more would defeat the purpose. We won't be in danger of running out of lithium anytime soon, it is one of the most common elements in the planet's crust and there is a lot of it in the ocean. Any shortage is going to be market driven which, while lithium is abundant, accessing much of it can be difficult and expensive. Remember that batteries are almost entirely recyclable though so you get a really long life out of it once you've mined it.