A near monopoly on rare earth mineral mining and refining certainly must play a part.
There's nothing preventing others from also processing concentrates (the post physical mining product).
The mining aspect is hardly monopolised either - much is sourced from Australia, South America, etc.
Put in an order, buy a 49% share of a resource company and now you're part of the action.
I'd also point out that the US has a large land area, mapped out by the USGS (below ground structures) and is capable of creating a large shallow lake area sealed off from deeper aquifers and able to be treated with neutralising agents.
On the scale of the market for global battery production the additional costs for doing it right are not punitive and arguably fall under the "national security" umbrella that see magnitude larger shovelfuls of money going to defence and traditional fossil fuel industries.