I have read Cohen's book, and various posts in this site; however, I'm still not convinced why we can't model a statistical mixture as a superpositions of pure states ?

For example, consider the Stern-Gerlach experiment; the book claims that when the atoms go outfrom the oven, we don't know their states; however, even when they pass from the apparatus, we still do now know the states of the particles coming from the $+z$ direction of the apparatus if we were to measure their $x$ component next. Of course, one can argue that, no we do know the state; it is $| +z \rangle$, but that state is composed of $|+x\rangle $ and $|-x\rangle$, so if were to measure their $x$ component, practically we just know that %50 percent of the atom will come out from the $+x$ direction.

Similarly, when we first pass the incoming beam coming out from the oven, we, statistically, knew that %50 percent of the beam was going to come out from $+z$ direction, so, in practice, I don't see any reason why can't we model the statistical mixture as a superposition of pure states.

For example, why don't we say that the state of the atoms in the beam coming out from the oven is $$\psi = 0.5 |+z \rangle + 0.5 |-z\rangle~?$$