Although rare earths aren't rare, they are rarely found in high enough concentrations to be mined profitably.
It also isn't a simple supply-demand story-- it takes over a decade to set up rare earth mines and refineries [1]. As the article mentions:
"China so completely dominates one key stage of the manufacturing process — converting the oxides to metals"
If China cut off U.S. access to rare earth magnets (as suggested by the article) or rare earth metals at other stages in the refining process, other countries could try to pick up the slack, but it would likely be devastating for a lot of companies for many years. Even much of the knowledge about how to refine the metals has been lost in the U.S. [2], which is partly why MP Materials (the RE mine in California) needed to have a partner in China.
[1]
"Rare metal mines (and supply chains) can take up to fifteen years from investment to production according to the U.S. government"
Abraham, David S.. The Elements of Power: Gadgets, Guns, and the Struggle for a Sustainable Future in the Rare Metal Age . Yale University Press. Kindle Edition.
[2]
"The lack of a knowledgeable workforce, even in countries with strong mining histories like Australia and Canada, will mean a slow development of local mines in Western countries and a growing reliance on mines developed in China, Russia, or Brazil, by Chinese, Russian, or Brazilian engineers. It also means that new young Western workers will have increasingly fewer mentors, crucial to these young scientists in developing their skills and providing them critical information."
Abraham, David S.. The Elements of Power: Gadgets, Guns, and the Struggle for a Sustainable Future in the Rare Metal Age . Yale University Press. Kindle Edition.