Part of the trouble with refining "rare earths" [1] is that the undesired residue (commonly called "slag") is radioactive and toxic. Smokestack emissions are also toxic enough that countries with pollution controls don't want them inside their borders. In the US, that means that every rare earth refinery becomes a SuperFund site [2].

China doesn't want to keep refining the metals - they want to move up the value chain by making things out of these metals. Instead of selling the refined neodymium & dysprosium for $50, they want to sell the electric motors that sell for $1,000.

Notes:

1 - They aren't rare at all, they're the bottom 2 strips/rows of the periodic table (of how it is most commonly displayed). Chemically, they're rather similar so the separation process is more complicated and annoying than, say, refining iron ore. Many people like to specifically exclude the actinides (the bottom row which includes uranium & plutonium) from the category "rare earth" because scary! radioactive! nuke! stuff! tends to divert discussion. https://en.wikipedia.org/wiki/Rare-earth_element

2 - A major problem with SuperFund sites is that every person/corporation who owned that land at any time is responsible for cleaning up the toxic waste. Just like asbestos waste. https://en.wikipedia.org/wiki/Superfund

Exactly! The problem with the way countries move up is to move the dirty industries to the up-and-coming countries. We need to properly pay for clean industry, but that conversation really never gets very far.