The only EV producer that's outright complained about battery availability is Tesla, and it's very easy to wonder how much of that was marketing spin to cover other production issues (and the accusation of the traditional car manufacturers that car production is hard and has huge startup hurdles), and maybe even PR/marketing spin to their own investors to explain insourcing battery production (building the "gigafactory") rather than partnering with an existing battery manufacturer.
Tesla probably has scaled faster than most other car companies, but yet it has also scaled much less comparably (to say Renault or VW just signing delivery or plant contracts in partnership with existing battery manufacturers rather than building their own factories from 100% "scratch").
I'd love to see a better analysis of the battery market right now. I'm betting the catch-22 that car manufacturers don't feel a lot of consumer demand for EVs has more on bottlenecking EV production than batteries in the current term.
Being at the 3rd place in the atomic table isn't a guarantee of availability - after all hydrogen is first, is 75% of the mass of the universe, and yet isn't directly available to us ! (at a low cost) And recycling batteries won't help much when the goal is to massively increase production...