Other issues
There are quite a few places in the US with significant deposits of rare earths - and some of those are in the "scoop up and start processing" range for accessibility. If someone is smart enough to stock a few months of rare earths to use while getting production online, a Chinese shutdown wouldn't do much more than raise prices.
There are also - in theory - some other options. You can recover a fair amount from the waste products left over from other mineral extraction processes. The "red mud" from bauxite processing, for example, can be used as a source of rare earths.
Just remember: "rare earth" elements aren't that rare - just hard to extract.