The mixed material problem is a glue problem

Two useful materials bonded together often make one useless one.

Almost every recovery process works on a single material at a time. Paper is pulped, and the fibres are separated out in water. Rigid plastics are shredded, washed and melted, sorted by type before they go anywhere near heat. Metals are melted. Each of these processes assumes the incoming stream is overwhelmingly one thing, and each of them is disrupted by the presence of anything else. So the moment two materials are bonded together tightly enough that they cannot be pulled apart, the combined object usually cannot enter any stream at all, because whichever process receives it will be contaminated by the half it was not built for.

Laminates are the classic case: a thin metallic or plastic film bonded to paper or board to give a barrier, a gloss or a stiffness that neither material offers alone. The result performs beautifully as packaging and is extremely difficult to unpick. The same problem appears wherever a plastic window is glued into a card sleeve, wherever a coating is applied to make board water resistant, and wherever an adhesive label is strong enough that it comes away with fibre attached. In every case, the bond that made the package work is the bond that ends its life.

The important point is that this is a design outcome rather than a consumer one. By the time a package is in your hands, the decision has already been made, and no amount of careful sorting on your part can unmake it. The tearing test people perform at the bin, picking at a corner to see whether a film will come away from a board, is an attempt to undo an engineering choice with fingernails. Whether it can be undone at all was settled at the drawing stage by someone weighing shelf life, printing quality and cost, quite possibly without ever being asked what should happen to the thing afterwards.