The moment they start doing this, other countries will restart their mines and the Chinese leverage (and profits) will evaporate. Rare earth minerals aren't that rare.
edit: latrare brings up another very important point. The patents surrounding the processes to refine RE materials. Take a look at who owns most of them...
Now, the irony is that if American companies want to start rare earth refining, then they must pay China for the use of those refinement patents.
Oh the irony.
Unless some American senator want to invalidate all of China’s patents on grounds of National Security. Oh the hypocrisy.