As far as I know only certain Nickel metal hydride batteries use rare earth metals. Unless you just meant "rare" as in uncommon. Nickel isn't particularly rare and can be recycled from batteries.

Nickel is quite rare in the sense that the volume available for us on earth isn't as robust as something like iron or lithium, which are extremely abundant.

Also Nickel is highly concentrated in set regions which also causes problems.

Nickel & Cobalt are the most expensive metals in NCA/NMC batteries which are very common today.

Nickel is 4x as abundant in the Earth's crust as lithium.

Peridotite, a mantle-derived ultramafic rock that's widely available around the world, can be 0.1-0.3% nickel. This rock has been considered for CO2 sequestration, as it has large amounts of olivine, which is one of the most easily weathered silicates for mineral carbonation. One concern of mineral carbonation is all the nickel that could be released.

Nickel is viable for mining in set regions as I've said before.

Although Li-ion batteries are named over lithium, they require quite a small amount of it compared to other materials. the bulk of the battery is the Cathode.

And the anode! Just responding since I work on Cu anode foils.

Cost and materials wise, Cathode are a larger portion.

Of course, Anodes are of extreme importance as well but Graphite\Silicon are more manageable than Nickel for example.

Anodes are more of an enabler for other things. for instance, anode almost solely dictates the charging rate of the battery. very important parameter for sure.

Neither nickel nor cobalt are rare earth metals.