I don't think it's trivially true in a technical sense at all. It certainly didn't have to be the case that all quantum systems exhibit entanglement. Physics could have been different, such that entanglement is a property only some systems can have. And that would lead to drastically different conclusions. In fact, treating entanglement as something special is, at least for me personally, the highly misleading concept.
When I was learning quantum mechanics, there were many concepts that confused and bothered me because people explained things in a loose and inaccurate way. Other incorrect statements that confused me for a long time include:
"Fermions cannot occupy the same quantum state. Bosons can."
"It's not possible to measure the position and momentum of a particle simultaneously."
And while not QM: "Mass can be converted to energy."
Essentially, schools teach the approximations first, which leads you to believe that reality works a certain way, and then you think of all sorts of situations where these rules lead to paradoxes and contradictions, and then you have to relearn everything the more accurate way, while trying to expunge the old, wrong stuff from your brain that likes to stick there.
In my undergrad, I thought of plenty of ways that "entanglement", based upon the way it was explained in class could be used to transfer information faster than the speed of light. The professor didn't have a rebuke against my argument (granted this was a chemistry professor, not a physics professor), and essentially it was because everyone was repeating wrong, catchy phrases to each other. I don't think "Entanglement is the fact that a multiparticle wavefunction cannot be decomposed" is a particularly complex idea to understand. Everything else kind of falls out of that statement.
In my opinion, the most accurate theory should be taught first, and then the less accurate approximations can be derived as limiting cases of the more accurate formulation. And if the math is too complex for the more accurate version, then clearly specify what fallacies or assumptions are being taken. One of my statistical mechanics books does this really, really well (the one by McQuarrie). Disclaimers are all throughout his book about how the equations apply to limiting cases and approximations, what those approximations are, and how using those approximations causes the result to differ from a more accurate theory.
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.
You have good points. However I'm going to be a little nitpicky here. Mixed states aren't exactly "real" in the sense that what is a pure state for Alice can be a mixed state for Bob (I can provide an example if you don't know what I mean). So in that case the "lack" of entanglement is simply due to ignorance of which pure state the system is actually in.
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).
(Sorry for continuing the rather long discussion, but I'd like to achieve some resolution here.)
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.
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.
The correct statement for the first one is that the wavefunction for a system of fermions must be antisymmetric under the exchange of any two particles (the wavefunction flips sign). For a system of bosons, it stays the same.
The correct statement for the third one is that energy is not a substance; it's a number that is conserved as a result of the invariance of physics under time translations (see Noether's theorem). Thus, mass has an energy associated with it (see stress-energy tensor), and a particle that has mass can be converted into other particles that do not have mass, but energy is just a property that remains constant (except under GR, but then that's a whole other can of worms).
When I was learning quantum mechanics, there were many concepts that confused and bothered me because people explained things in a loose and inaccurate way. Other incorrect statements that confused me for a long time include:
"Fermions cannot occupy the same quantum state. Bosons can."
"It's not possible to measure the position and momentum of a particle simultaneously."
And while not QM: "Mass can be converted to energy."
Essentially, schools teach the approximations first, which leads you to believe that reality works a certain way, and then you think of all sorts of situations where these rules lead to paradoxes and contradictions, and then you have to relearn everything the more accurate way, while trying to expunge the old, wrong stuff from your brain that likes to stick there.
In my undergrad, I thought of plenty of ways that "entanglement", based upon the way it was explained in class could be used to transfer information faster than the speed of light. The professor didn't have a rebuke against my argument (granted this was a chemistry professor, not a physics professor), and essentially it was because everyone was repeating wrong, catchy phrases to each other. I don't think "Entanglement is the fact that a multiparticle wavefunction cannot be decomposed" is a particularly complex idea to understand. Everything else kind of falls out of that statement.
In my opinion, the most accurate theory should be taught first, and then the less accurate approximations can be derived as limiting cases of the more accurate formulation. And if the math is too complex for the more accurate version, then clearly specify what fallacies or assumptions are being taken. One of my statistical mechanics books does this really, really well (the one by McQuarrie). Disclaimers are all throughout his book about how the equations apply to limiting cases and approximations, what those approximations are, and how using those approximations causes the result to differ from a more accurate theory.