The density matrix is as real as the wave function when it comes to describing the corresponding subsystem. In the situation I was sketching, there is no measurement, no collapse, and the "measurement problem" does not play a role. Here is a concrete example: Suppose that you have three spins that are in a superposition of |000> and |111>. Alice has one the spins, Bob the other spin, and the third one belongs to the environment. The reduced state of any two of the three spins is NOT entangled. Therefore, Alice and Bob which will not be able violate any Bell inequality, win a CHSH game, distill Bell pairs, etc. if they only control two of the three spins. It is irrelevant that Alice is entangled with the joint system of Bob and Eve.
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.
> (And if you're referring to a direct tensor product of two single particle states, I'm still considering that an entangled state since it can unitarily evolve out of that configuration into one that can't be written as a tensor product)
This reasoning does not make sense. The notion of entanglement is not invariant under "global" operations, nor should it be.
The point is that entanglement always refers to an a priori choice of subsystems (say, Alice and Bob). This is the part that makes the phenomenon non-trivial. If there are other systems around (say, Eve the environment) then the joint state of Alice and Bob will be usually be mixed as a consequence of for the "trivial" reason that we discussed in the previous posts (to adapt a famous saying, almost all components of a quantum system are always mixed ;-). There is nothing "unreal" about mixed states, and not all mixed states lack entanglement. However, for mixed states, being entangled is no longer the generic behavior. The unavoidable interactions with the environment are the reason why it is hard to maintain entanglement between subsystems.
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).
I guess I was being generous when interpreting your statement that "all components of a quantum system are always entangled". For pure states, such a statement is trivially true in the sense any kind of interaction in the Hamiltonian will generically create entanglement.
For mixed states, your statement is of course false. It is unfortunately often the case in the real world that two subsystems are in a non-entangled quantum state.
Regarding your general theme: I understand that confusion can arise if seemingly informal language is taken verbatim as a formal statement. I do not think that the solution is to abolish the former, which can be extremely efficient to reason in and communicate with (you gave some examples in your post), but rather to educate on the interpretation. It is unfortunate that this is not always done, as your experiences suggest.
I would argue that using the term "entanglement" to refer to the situation where "the entanglement of the system is much more apparent" or "tensor products are not a good approximation anymore" is completely reasonable. In contrast, when you say that "all components of a quantum system [that appears in nature] are always entangled" then this is trivially true in a technical sense, but highly misleading. To produce and stabilize non-trivial amounts of entanglement in a way that it can be harnessed for quantum information processing is certainly an interesting and highly non-trivial task which people are spending lots of effort on.
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.