Chile would certainly benefit from having more battery manufacturing in its economy. They've recently joined the short list of countries where solar power is more than 10% of consumption — growing fast — and they still generate and consume much
less total electricity than a typical developed country. This implies that they're going to end up with a very large solar fraction, and a need for stabilization.
But you can't just make a battery out of lithium. You also need graphite and either nickel or iron phosphate. And you need some fine chemicals (the electrolyte). The good news is that Chile does produce some iron and phosphorus. But I don't think they currently produce much in the way of graphite or organic chemicals. I'm not sure, either, if they can process their iron and phosphorus ore into battery-grade iron phosphate. So it's not just "refining lithium". Lithium carbonate of commerce is already pretty pure. This isn't petroleum we're talking about.
The danger is trying too hard to achieve autarky. The CPTPP (of which Chile is a member) at least in theory does have existing sources for those other components. But it seems like the Boric administration is skeptical of cooperation. Environmentalists are another potential barrier, since anything involving chemical engineering tends to get them riled up, justified or otherwise.
By the way, I think you mean "downstream" when you say "upstream". Mines are generally the proverbial headwaters.