Rare Earths aren't rare in the "there is a small supply" sense, but in the "very dilute" sense. Rare earths don't concentrate into ores the way that say copper does. Rare earth deposits are just places where you happen to have 300 ppm instead of the crust average of 220 ppm.

The only way to mine rare earths is to just process massive quantities of earth. Typically this is done as part of another mining operation, like mining nickel. It's labor intensive and requires nasty chemicals. Places with cheap labor, weak environemental regulations, and extremely large scale mining operations that they are going to be operating anyways are always going to be able to produce the cheapest rare earths. It's very easy to see why China naturally dominates the market.