They are rare in the sense that they tend to be extremely diffuse, rather than having nice, massive veins of concentrated ore or elemental metal like you might find for copper or bauxite.
Since they tend to be diffuse, mining them requires disrupting significant volumes of earth and rock, plus the chemicals needed to separate them out of the less interesting material that gets dug up.
The cheapest way to do that is to strip mine large tracts of land and not reclaim or treat any water used in the process, which will likely be full of heavy metals and other chemicals.
If we don't like how other countries do it, we have to be willing to do it ourselves, which means years delayed supply chains (basically every mine in the US is protested and delayed through the legal system) and higher prices for the refined materials (it costs more to do it right).
It's called "liquid-liquid extraction" [1] and requires crushing the rock and mixing it with an extractant (see D2EHPA [2] which is also used in uranium extraction) into a nasty acidic slurry. It is then separated into a aqueous layer containing the waste and a nonpolar solvent that strips the rare earth elements bound up with the extractant. All the different rare earth elements then have to be separated out of the nonpolar solvent using even more toxic chemicals, each of which leaves behind a different hazardous waste.
Fun(?) fact: This process looks a lot like the alkali extraction process used to make cocaine, DMT, and a variety of other drugs.
Since they tend to be diffuse, mining them requires disrupting significant volumes of earth and rock, plus the chemicals needed to separate them out of the less interesting material that gets dug up.
The cheapest way to do that is to strip mine large tracts of land and not reclaim or treat any water used in the process, which will likely be full of heavy metals and other chemicals.
If we don't like how other countries do it, we have to be willing to do it ourselves, which means years delayed supply chains (basically every mine in the US is protested and delayed through the legal system) and higher prices for the refined materials (it costs more to do it right).