There is a metric sheet-tin of sodium in the concentrated brine...

There is a metric sheet-tin of sodium in the concentrated brine even by the time it goes to the lithium plant, the lithium is precipitated from the brine with sodium carbonate, giving a lithium carbonate product.  Thus lithium carbonate as a traded product was not because of market demand for lithium carbonate but because of the necessity to seperate the lithium from the sodium.  It's not entirely easy to prepare lithium chloride from the brine without a total rethink of the whole process.