We absolutely could mine and refine the stuff. It is everywhere. The "problem" is that the refining process is horribly toxic and the residue/slag is toxic and radioactive. The last refinery that had been located in the US is now a SuperFund site. Every country would prefer that the refining takes place in China because that way the pollution & hazardous waste remains in China.

The anger is that China is being "uppity" by wanting to make the things (like motors) out of the "rare earth" elements instead of being the colony that supplies raw materials to the Empire/colonizer. This was one of the complaints leading to the Revolution in 1776 - only raw materials could be shipped to England and all finished goods could only be shipped from England. The colonies were forced to remain at the bottom of the economy.

China wants "a seat at the table". Western countries are unwilling to let that happen.