'Physical realizability' in the Symmetrization Postulate

I have two questions about the $Symmetrization Postulate$:

In a system with $N$ identical particles, physical states aren’t arbitrary states in $V^{otimes n}$. Rather, they’re totally symmetric (belong to $Sym^NV$), in which case the particles are said to be bosons, or they’re totally anti-symmetric (belong to $Anti^NV$), in which case they’re said to be fermions. (From MIT OpenCourseWare.)

First, it is said that a postulate is something technically can’t be derived. In this case, is it because this postulation gives a statistical definition of bosons and fermions, which is something that more advanced theories based on so that it cannot be proved? Second, I’ve seen a Venn diagram showing the relations between $Sym^NV$, $Anti^NV$, and the total space $V^{otimes N}$. When $N = 2$, the union of those two subspaces is just the total space. However, when $N$ is greater or equal to 3, there’s something else in $V^{otimes N}$ which lies in neither $Sym^NV$ nor $Anti^NV$, those states are called partial-symmetry or partial-antisymmetry states, which could be found through Young tableaux. Thus I’m wondering since those partial (or can I replace ‘partial’ by ‘mixed’?) (anti-)symmetry states are not physically realizable, why are we interested in figuring them out, and what’s their significance?

Thanks!