Well, not at the promised rate, because they're not 1000x as efficient (which they would be promised in article upon article over 4 decades and compounded).
The other issue is recyclability. Modern li-ion batteries are basically superglued together in order to maximize energy density. A pack designed with recycling in mind will have to be more easily taken apart and likely significantly heavier.
My understanding is currently they just grind up the old batteries into a kind of slurry that then goes through a uniform series of processes to recover the various components.
A process that has both huge losses and requires enormous amounts of energy.