To say that we should do better and bring the environment back into the picture is missing the point if we are interested in the correlations between Alice and Bob. These do exclusively depend on their joint state (mixed or not).
By "real", I mean that the concept of a density matrix is derived completely on top of the postulates of quantum mechanics in combination with the Born rule. There's nothing fundamental about mixed states; you can also have classical mixed states (e.g. deriving thermodynamics from classical statistical mechanics). It's essentially just taking the postulates for pure states and applying a layer of statistics on top of it. I believe it was von Neumann that originally did this? In other words, the density matrix formulation does not add any additional predictive capability to physics that the original QM from the 1920s did not already provide. It's just a more convenient tool for connecting QM to experimentally realizable systems. Do you disagree with this?
When you lose entanglement due to decoherence (specifically, the off-diagonal terms of the density matrix approaching zero), these correlations are lost because you're essentially performing a measurement. But they still exist in the whole Alice + Bob + you and your measurement device system! But then, this starts treading into the discussion of the whole unsolved measurement problem which I kind of wanted to lurk around, since no one ever gets anywhere with those discussions.
Indeed your last point ("To say that we should do better and bring the environment back into the picture is missing the point") is essentially the whole picture I'm focusing on. Perhaps my background with quantum chemistry has slanted the way I explain things on here, because you're never collapsing these systems when you perform simulations of them to calculate their properties.
Again, the basic point is that the notion of entanglement refers to a choice of subsystems. Your statement that "All components of a quantum system are always entangled" is either trivializing the discussion or demonstrably false.