This is not going to happen because, as you point out, lithium mining will only expand if prices climb higher to make otherwise unprofitable reserves profitable. And since raw materials now dominate the cost of batteries, this is going to increase the cost of batteries. There's no having your cake and eating it too: in order to increase lithium mining capacity, battery cost is going to have to grow, not shrink.
You edited your post after I replied, so I'll have to edit in response:
There's no one "difficulty of extraction" factor for each commodity. The reality is that there's a diverse variety of reserves all of which are easier or harder to exploit, even for the same commodity. "The price is determined by how hard it is to extract" is at best a huge simplification.
Higher commodity prices make it viable to extract the more inaccessible reserves, but those reserves will only be profitable so long as prices remain high.
> If lithium was profitable to mine for $X/kg for many years, then that is the price it will naturally return to.
Nope! This is completely wrong. Commodities don't "naturally return" to any price. Prices is a result of supply and demand. It could be we find some other battery chemistry that blows lithium out of the water. In that case the cost of lithium will probably collapse well below $X/kg. Conversely, if countries start to try and provision significant amounts of grid battery storage, the costs will grow even higher as more and more inaccessible reserves need to be exploited to supply market demand.
There is no "natural" price of commodities, whatsoever. If the demand for lithium is going to rise and keep rising, then the cost will rise and keep rising unless some breakthrough makes it way more efficient to mine. Given the fact that we've been mining for centuries and we have huge demand for minerals other than lithium, I'm not optimistic on a 100x improvement in mining efficiency.