I am trying to understand the necessity of density matrices and the notion of "mixed states" in quantum mechanics (I read all the other posts about this, I promise).
As far as I understand, one could motivate these notions as follows:
Let $H_1$ and $H_2$ be Hilbert spaces of two quantum systems $S_1$ and $S_2$. Moreover, let $H_1 \otimes H_2$ be the Hilbert space of the composite system $S$. Now, let $x \in H_1 \otimes H_2$ be a pure state, by which I mean just some arbitrary (unit) vector. If $x$ is entangled (i.e. cannot be written as $x=x_1\otimes x_2$) then we cannot in a reasonable way reduce $x$ to pure a state (again, read (unit) vectors) in $H_1$ or $H_2$. A way out of this is to introduce density matrices in order to relax the "state"-notion just enough that the "reducing to subsystem" is well-defined by taking the partial trace.
My question is now: Why would we want to "reduce" to $S_1$ or $S_2$ in the first place? If we want to measure some $S_1$-observable $A$ in $S$, we could just apply $A \otimes I$ to $x$ itself which should give the same expectation values etc., right? Of course, I'm hiding here that $x$ itself could be a density matrix but I feel like my argument works inductively. Are there other benefits of introducing "mixed states" which I am not seeing?