Despite their name, Rare Earth metals are not rare... Lanthanum is the only rare-earth used in batteries (but in Nickel, not Lithium chemistries), and it's 80% as prevalent as Copper in the Earth's crust. Neodymium, most commonly used in magnets, is slightly more common again. The actually-rare rare metals have no automotive or battery applications that demand large quantities.
The other battery metals are not rare by any measure: Lithium is one of the most abundant metals on Earth, as you'd expect from its low atomic number. It can be commercially extracted from a variety of sources, from salt-flats to sea-water - one problem is that there's almost no infrastructure for recovery and recycling of lithium, but this will come as demand increases. Cobalt is mined with copper, and was until recently a low-value byproduct of copper mining - it's not in short supply, although there are ethical concerns about the treatment of miners in the countries with the largest deposits.. Manganese is twenty times more common than copper, and is mined with Iron.
Every claim that we're running out of Rare Earths is accompanied by the disclaimer "unless new supplies are found". Until the last twenty years or so, there really wasn't much demand for these metals, so there was only a limited number of supplies, and no incentive to find more. Nobody went looking for oil in Saudi Arabia until oil was something that was worth money...
Extraction of rare-earths is an economic proposition. When prices increase, supplies will become viable.
I'd rather not discuss theories on climate change here. I don't agree with your analysis - it ignores the huge difference in the atmospheric lifetime of CO2 versus water vapor for one - but this isn't the place for an argument on this subject.