As it turned out, that is almost exactly what happened. The bubble of high prices made it economically viable for other rare earth mines to open, and they did. The mothballed Mountain Pass mine came back online in 2012, ramping up to full production in 2015, and in May 2011 the Australian miner Lynas Corporation started production at its Mount Weld rare earth mine. Since then, Lynas has built a refining facility in Malaysia to process rare earth products.
So long as every technology company wants to wait a couple years before making product again, everything will work out fine for capitalism.