Rare earth metals are, despite the name, not actually all that rare. They're fairly abundant in the Earth's crust and mantle.

The reason they're more expensive than, say, iron, isn't that they're rare but they're rather difficult to purify from each other. They share the same outer shell electrons (s and d), and what separates them from each other is their f-shell electrons. Those, however, are buried inside the other shells. So their differences are largely driven by differences in atomic radius, differences which are small in chemical effect.

Which is all to say that the cost of producing them wouldn't change too much in that scneario, even if you had access to a trillion tons of ore for free.