Lithium is valuable because extracting it from naturally occurring salt deposits is a lot of work. These salt deposits are mostly not Lithium. We are talking trace amounts here. To extract it, you have to process vast amounts of brine. Boil of the water, separate it from other materials, etc. That's a lot of work and energy. Recycling a battery with high concentrations of Lithium is probably a lot less work. For that reason, lithium is a lot more expensive than other materials.
Melting old ice cars to recover the steel is already a thing. The steel has a value. All you need to do is melt it and reuse it. So, we already have existing practices for recycling old vehicles and companies specializing in that. The only thing that changes is that those companies will be dealing with very valuable batteries as well. Lithium is of course quite a bit more lucrative than steel. If it works for steel, it's going to work a lot better for lithium. Because it has a lot more value.