Yeah, not really. Just like with "oil reserves", there's a common misconception that we'll "run out" when they are depleted. However, "reserves" are just those resources that can be extracted economically at today's prices. When demand increases, prices rise and reserves "magically" increase.
That's exactly what will happen here. "Rare-earth materials" aren't actually rare in the sense that precious metals are rare. They're just uncommon compared to other minerals. If prices rise, eventually China will decide to increase exports and other countries that aren't literally warzones (like the DRC) will chime in.
So, we have ten years until those batteries will become somewhat more expensive, making research into alternatives more promising. There is no urgency here.
Then of course there's the questionable idea that electric motors will replace combustion engines because we're "running out of oil", which is why we supposedly need all these batteries in the first place. My bet is that synthetic fuels will become profitable far earlier.