Lithium is a valuable resource. EV batteries have lots of it. Therefore EV batteries are valuable even after they stop working. 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…
A Tesla battery weighs about 500 kg and has about 10kg of Lithium (2%). The Lithium is worth about $130, whereas the battery costs more than $10,000. It's difficult to extract the trace elements of lithium from the thousands of individual cells honeycombed together in the battery, so there is no known way of getting at that Lithium via processes that are even comparable to the value of the lithium recovered. That's why no one does it.
A car, on the other hand, is 55% steel by weight, and there are relatively straightforward ways of getting that steel (no, the car is not "melted down". The steel components are melted after they are removed from the car). That's why steel in cars is recycled but Lithium in batteries is not.
The issue with recycling is the cost of extraction -- yes, catalytic convertors and old computers contain trace elements like platinum (maybe $2 per catalytic convertor) but extracting it is far too expensive. So merely the presence of an expensive element in a manufactured good does not mean it is economic to extract the element. In the vast majority of situations, it's not.