Anything made of steel or aluminum is recyclable because they can just melt it down and easily separate the metals, but the carbon lining and anything nonmetal is basically slag afterwards. Aluminum, electrolyte, and random atoms seep in everywhere and destroy it.
The smelting process I described above is actually the more expensive process to used to produce aluminum from raw bauxite. Recycling aluminum is cheaper and a significant fraction of the world’s aluminum produced every year is from recycled feedstock (over two thirds in the US, last I checked). Same goes for steel and most other metals.
I'm sure, like any metal at an industrial scale, it is profitably recyclable. But that is beside the point. This is akin to asking: "My car's engine just threw a rod and is seized. Is it recyclable?" Hopefully you see in this analogy that the car (engine) costs way, way more than the sum of its parts (the constituent metals).
The smelting process I described above is actually the more expensive process to used to produce aluminum from raw bauxite. Recycling aluminum is cheaper and a significant fraction of the world’s aluminum produced every year is from recycled feedstock (over two thirds in the US, last I checked). Same goes for steel and most other metals.